Martin Pölcher

Meike Eckhardt

Christoph Coch

Matthias Wolfgarten

Kirsten Kübler

Gunther Hartmann

Walther Kuhn

Christian Rudlowski
email: christian.rudlowski@ukb.uni-bonn.de

Sorafenib in combination with carboplatin and paclitaxel as neoadjuvant chemotherapy in patients with advanced ovarian cancer

Keywords: Sorafenib, Ovarian cancer, Neoadjuvant chemotherapy, Carboplatin, Paclitaxel

published or not. The documents may come

Introduction

Epithelial ovarian cancer is the leading cause of death from gynecologic malignancies [START_REF] Jemal | Cancer statistics, 2006[END_REF]. Due to nonspeciWc initial symptoms, most cases of ovarian cancer are diagnosed at an advanced stage [START_REF] Cannistra | Cancer of the ovary[END_REF]. The current standard therapy for these patients is a combination of cytoreductive surgery followed by platinum-and taxane-based chemotherapy. Patients with International Federation of Gynecology and Obstetrics (FIGO) stage IIIC and IV disease have an unfavorable prognosis with 5-year survival rates varying from 19 to 33% depending on resectability and response to chemotherapy [START_REF] Heintz | Carcinoma of the ovary. FIGO 6th annual report on the results of treatment in gynecological cancer[END_REF]. In ovarian cancer, numerous studies demonstrated the feasibility of preoperative chemotherapy in order to down-stage tumor load and to evaluate treatment response [START_REF] Vergote | EORTC-GCG/NCIC-CTG randomised trial comparing primary debulking surgery with neoadjuvant chemotherapy in stage IIIC-IV ovarian, fallopian tube and peritoneal cancer (OVCA)[END_REF][START_REF] Pölcher | Neoadjuvant chemotherapy with carboplatin and docetaxel in advanced ovarian cancer-a prospective multicenter phase II trial (PRIMOVAR)[END_REF]. Given the eYcacy plateau reached in the treatment of ovarian cancer with standard cytotoxic therapy, recent strategies have focused on molecular targets that inhibit mediators of angiogenesis.

123

Sorafenib is an oral multi-kinase inhibitor targeting both tumor cells and the tumor vasculature [START_REF] Wilhelm | BAY 43-9006 exhibits broad spectrum oral antitumor activity and targets the RAF/MEK/ERK pathway and receptor tyrosine kinases involved in tumor progression and angiogenesis[END_REF]. As inhibitor of signal transduction, sorafenib acts on the Raf/MEK/ERK pathway [START_REF] Thompson | Recent progress in targeting the Raf/ MEK/ERK pathway with inhibitors in cancer drug discovery[END_REF] at the level of Raf kinase [START_REF] Caraglia | Targeting Raf-kinase: molecular rationales and translational issues[END_REF] and on tyrosine kinases vascular endothelial growth factor receptor-2 (VEGFR-2) and platelet-derived growth factor receptor (PDGFR-). Sorafenib demonstrated single-agent activity in multiple phase I/II studies, and inhibited tumor growth in patients with refractory solid tumors of diVerent sites. In addition, combination of chemotherapy and sorafenib showed partial response or stable disease.

The clinical beneWt of sorafenib has been demonstrated in a randomized, double blind, placebo-controlled phase III study in patients with renal cell and hepatocellular carcinoma, leading to an approval of this medication [START_REF] Escudier | Sorafenib for treatment of renal cell carcinoma: Wnal eYcacy and safety results of the phase III treatment approaches in renal cancer global evaluation trial[END_REF][START_REF] Llovet | Sorafenib in advanced hepatocellular carcinoma[END_REF].

The potential activity of sorafenib in ovarian cancer is supported by preclinical studies [START_REF] Gianpaolo-Ostravage | Anti-tumour eYcacy of the orally active raf kinase inhibitor BAY 43-9006 in human tumour xenograft models[END_REF]. Furthermore, patients with recurrent ovarian cancer were treated within a phase II-study [START_REF] Welch | Phase II study of sorafenib (BAY 43-9006) in combination with gemcitabine in recurrent epithelial ovarian cancer: a PMH phase II consortium trial[END_REF] and within several phase I-studies in solid tumors [START_REF] Strumberg | Pooled safety analysis of BAY 43-9006 (sorafenib) monotherapy in patients with advanced solid tumours: is rash associated with treatment outcome?[END_REF].

This study was designed to assess eYcacy of combination therapy of sorafenib and standard chemotherapy within the neoadjuvant setting. A special focus was put on preoperative response evaluation and on translational research projects for comparing pre-and posttreatment samples.

Patients and methods

This study included patients with histologically conWrmed FIGO stage IIIC and IV epithelial ovarian cancer and an ascites volume of 500 ml or more. The main exclusion criteria were mucinous cell type, debulking procedures during initial surgery, and signs of bowel obstruction.

Patients received carboplatin at area under the curve 5 and paclitaxel 175 mg/m 2 intravenously in a three-week schedule and sorafenib was given 400 mg twice daily (bid) orally. The treatment schedule is shown in the table. The primary endpoint was Progression-free survival. A clinically meaningful improvement was deWned as a hazard ratio of 1.5 of historic treatment (PRIMOVAR study [START_REF] Pölcher | Neoadjuvant chemotherapy with carboplatin and docetaxel in advanced ovarian cancer-a prospective multicenter phase II trial (PRIMOVAR)[END_REF]) over sorafenib treatment. A total of 102 patients should be enrolled [START_REF] Dixon | Sample size considerations for studies comparing survival curves using historical controls[END_REF] (Table 1).

Adverse events were recorded and graded according to the National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAEv3.0). All patients provided informed consent, and the protocol was approved by the local ethic committee.

Results

From July through November 2007 four patients were enrolled. A diagnostic laparoscopy was performed to record the peritoneal spread and to conWrm the diagnosis by taking biopsies of the peritoneal surfaces. After preoperative treatment and cytoreductive surgery, all patients were excluded from protocol due to severe (grade 3/4) toxicities. The study was stopped on the recommendation of an independent data safety monitoring board.

Brief description

The Wrst patient was 72-year-old woman with a past medical history of diabetes complicated by a diabetic angiopathy and status post coronary artery bypass surgery 15 years ago. She presented with a stage IV disease and a symptomatic pleural eVusion requiring multiple drainage. At the time of admission, cardiac examination was within normal limits. After the Wrst cycle of chemotherapy and 2 weeks of study medication with sorafenib, she was admitted to the hospital with signiWcantly impaired performance status, cardiac failure, edema and severe changes in electrolyte metabolism. The patients were subsequently admitted to the intensive care unit for close observation and diuretic treatment. The length of hospital stay was 7 weeks. She was excluded from the protocol and chemotherapy was continued with single agent carboplatin therapy. After a total of two preoperative cycles of chemotherapy, serum CA 125 levels decreased from 3,600 to 150 U/ml. Cytoreductive surgery was performed without complications (no macroscopic residual disease) and systemic treatment was completed with four additional cycles of postoperative single agent carboplatin chemotherapy. Recurrence occurred after 13 months, and the patient died 21 months after initial diagnosis.

The second patient, 72 years old, presented with a stage IV disease with a malignant pleural eVusion. Due to skin The third patient was 64 years old and presented with stage IIIC disease and a performance status ECOG 0 with an uneventful past medical history. Sorafenib was reduced after the Wrst cycle due to skin toxicity grade II as recommended by the protocol. CA 125 baseline level dropped from 2,000 to 105 U/ml. All visible tumors were reduced to less than 1 cm. In the early postoperative course, she had a symptomatic posterior myocardial infarction. The conducted PTCA and anticoagulation treatment caused hemorrhage in the abdomen and necessitating re-laparotomy. She was excluded from the protocol and had a progression-free survival of 29 months after four postoperative cycles of single agent carboplatin. On last follow-up, there was no evidence of disease.

The fourth patient was 34 years old and presented with stage IIIC disease. After Wrst cycle of therapy, dose reduction of sorafenib to 50% was required as skin toxicity grade II and gastrointestinal toxicity grade II was documented. CA 125 serum level decreased from 400 U/ml initially to 275 U/ml. However, on laparotomy peritoneal carcinomatosis appeared more pronounced compared to initial laparoscopic Wndings. Cytoreductive surgery was performed, making small and large bowel resection and a partial bladder resection necessary (residual tumor: •1 cm). An anastomotic leak of the rectosigmoid was diagnosed 5 days after surgery. Re-laparotomy was performed and had to be repeated after another week on account of peritonitis, leading to extensive resection of small bowel. (temporary abdominal closure and short bowel syndrome). Patient was excluded from the protocol, and chemotherapy was conducted with topotecan. She had an isolated recurrent tumor at the abdominal wall, which was treated with radiation therapy. Today she is alive 26 months after initial diagnosis and receives treatment of recurrent disease.

Discussion

We report a phase II-trial on oral sorafenib in combination with carboplatin and paclitaxel as neoadjuvant chemotherapy in the treatment of patients with advanced ovarian cancer. After enrollment of four patients, all patients were excluded due to adverse events. Three of them suVered lifethreatening events. Due to toxicity, dose reduction of the study medication sorafenib was necessary in all patients. Two patients were classiWed to have primary progressive disease. Based on these preliminary Wndings, an intolerable increase in toxicity and morbidity of oral sorafenib in combination with standard carboplatin/paclitaxel could not be excluded.

Most trials support sorafenib 400 mg bid as the recommended dose for combination therapy [START_REF] Strumberg | Phase I clinical and pharmacokinetic study of the Novel Raf kinase and vascular endothelial growth factor receptor inhibitor BAY 43-9006 in patients with advanced refractory solid tumors[END_REF]. Sorafenib as a single agent was generally well tolerated; most adverse events were mild to moderate in severity. Frequently reported drug-related adverse events at any grade included fatigue (40%), anorexia (35%), diarrhea (34%), rash/desquamation (27%), and hand-foot skin reaction (25%). Drug-related toxicities of sorafenib have been also reported in phase I and II trials for several coadministered drugs, including oxaliplatin, gemcitabine and carboplatin/paclitaxel [START_REF] Welch | Phase II study of sorafenib (BAY 43-9006) in combination with gemcitabine in recurrent epithelial ovarian cancer: a PMH phase II consortium trial[END_REF][START_REF] Takimoto | Safety and anti-tumor activity of sorafenib (Nexavar) in combination with other anti-cancer agents: a review of clinical trials[END_REF]. Recently, within a double blinded, placebocontrolled phase III-trial in patients with advanced melanoma, 270 patients were randomly assigned to receive intravenous paclitaxel (225 mg/m 2 ) and carboplatin at area under the curve 6 according to a 21-day cycle and either placebo or oral sorafenib 400 mg bid on days 2-19 [START_REF] Hauschild | Results of a phase III, randomized, placebo-controlled study of sorafenib in combination with carboplatin and paclitaxel as second-line treatment in patients with unresectable stage III or stage IV melanoma[END_REF]. Though dermatologic grade 3 adverse events were signiWcantly increased in patients treated with sorafenib, hematologic and nonhematologic adverse events did not diVer signiWcantly between the placebo group and the patient group with sorafenib. The treatment was considered generally feasible particularly no continuous serious adverse events emerged within this treatment schedule, which represents a similar one compared to our study. Dose reduction was performed in 33% (sorafenib group) and 10% (placebo group). The combination therapy with oral sorafenib in this trial did not increase response rates or progression-free survival signiWcantly over the standard treatment arm.

The serious adverse events in the current study raised the essential question, whether the neoadjuvant treatment could contribute to these side eVects and whether this treatment after cytoreductive surgery could be less toxic. Ovarian cancer is an intraperitoneal disease with large tumor masses at presentation, especially in preselected advanced ovarian cancer patients with large volume ascites. Apoptotic and necrotic eVects of antineoplastic therapy leads to degradation of large tumor cell masses, which may mediate an increased toxicity.

The combination of anticancer therapy and cytoreductive surgery might enhance postoperative morbidity and/or mortality. Targeting the VEGF pathway could impair wound healing and promote anastomotic leaks after bowel surgery. This was considered in the present treatment schedule and preoperative medication of sorafenib was stopped at least 10 days before surgery.

In two patients, progressive disease was diagnosed after neoadjuvant treatment: surgical exploration showed an 123 increased number of peritoneal tumor implants. This was an unexpected Wnding, because in 43 patients treated at our institution within the PRIMOVAR trial, none of the patients were diagnosed with primary progressive disease [START_REF] Pölcher | Neoadjuvant chemotherapy with carboplatin and docetaxel in advanced ovarian cancer-a prospective multicenter phase II trial (PRIMOVAR)[END_REF].

Recent data demonstrates that some chemotherapeutic drugs can cause simultaneously host-mediated counterregulatory responses from the bone-marrow resulting in tumor angiogenesis and vasculogenesis. This mobilization eVect in circulating endothelial progenitor cell levels may facilitate tumor cell repopulation during the common time intervals between the individual chemotherapy cycles [START_REF] Shaked | Rapid chemotherapy-induced acute endothelial progenitor cell mobilization: implications for antiangiogenic drugs as chemosensitizing agents[END_REF]. Furthermore, Paez-Ribes et al. [START_REF] Paez-Ribes | Antiangiogenic therapy elicits malignant progression of tumors to increased local invasion and distant metastasis[END_REF] reported that VEGF inhibition might play a role in regrowth of tumor, potentially inXuencing the progression of metastatic disease. Pretreatment of healthy mice with VEGF inhibitor sunitinib prior to intravenous inoculation of breast cancer cells led to more aggressive metastasis as well as a brief treatment after resection of the primary tumor. The underlying mechanisms remain to be explored but a withdrawal of anti-VEGF therapy might promote vascular recovery and a rapid tumor re-growth [START_REF] Ebos | Accelerated metastasis after short-term treatment with a potent inhibitor of tumor angiogenesis[END_REF]. A short time treatment in a neoadjuvant setting may lead accidentally to a stimulation of at least parts of the tumor or may promote metastasis. Prolonged intervals of treatment-discontinuation of antiangiogenic agents may also play a relevant role in the postoperative treatment of patients with residual disease or of patients with recurrent disease. Recently, several phase II studies were initiated analyzing a combination of sorafenib and carboplatin/paclitaxel for patients with primary ovarian cancer or recurrent disease and a combination of sorafenib and topotecan for patients with platinum resistant disease.

In conclusion, the current study was stopped because of serious adverse events. Considering the results in a recent reported trial [START_REF] Hauschild | Results of a phase III, randomized, placebo-controlled study of sorafenib in combination with carboplatin and paclitaxel as second-line treatment in patients with unresectable stage III or stage IV melanoma[END_REF], the Wndings of our study is probably based on random events. Future studies will be required to adjust the signiWcance of antiangiogenic and tyrosinekinase inhibitory therapy in patients with ovarian cancer and the eYcacy of concomitant treatment schedules or maintenance therapy.

ConXict of interest statement Christian Rudlowski: Research Funding (Bayer HealthCare AG, Deutschland). All other authors indicated no potential conXict of interest.

Table 1

 1 Treatment plan of neoadjuvant combination therapy with carboplatin/paclitaxel and sorafenib d day, OP operation, M maintenance , the sorafenib dose was reduced to 50%. After 2 cycles of treatment, CA 125 serum level decreased from baseline 2,156 to 544 U/ml. However, compared to the patterns of spread documented during laparoscopy, we found an increased number of peritoneal tumor sites at cytoreductive surgery. Residual tumor size: >1 cm. Suggesting platinum refractory tumor, further treatment proceeded with topotecan. Patient recurred after 12 months and died 24 months after initial diagnosis.

	Cycle	Medication	Length
	number		(days)
	1	Carboplatin and paclitaxel d1	21
		sorafenib from d2 to d19
	2	Carboplatin and paclitaxel d1	21
		sorafenib from d2 to d11
		(rest sorafenib for 10 days before surgery)
	OP	Cytoreductive surgery
	Recovery	Recovery without chemotherapy	14
		or sorafenib treatment 14 days
	3	Carboplatin and paclitaxel d1	21
		without administration of sorafenib
	4-6	Carboplatin and paclitaxel on d1	21
		sorafenib from d2 to d19
	M1-M14	Maintenance phase with
		sorafenib single agent


Jean Dolbeault

SOBOLEV AND HARDY-LITTLEWOOD-SOBOLEV INEQUALITIES: DUALITY AND FAST DIFFUSION

Keywords: March 5, 2011 Sobolev spaces, Hardy-Littlewood-Sobolev inequality, logarithmic Hardy-Littlewood-Sobolev inequality, Sobolev's inequality, Onofri's inequality, Gagliardo-Nirenberg inequality, extremal functions, duality, best constants, stereographic projection, fast diffusion equation, extinction. -AMS classification (2010). 26D10, 46E35, 35K55

In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension d ≥ 3. The main consequence is an improvement of Sobolev's inequality when d ≥ 5, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension d = 2, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equation.

Introduction

In dimension d ≥ 3, it is well known since E. Lieb's paper [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF] that Hardy-Littlewood-Sobolev inequalities are dual of Sobolev's inequalities. We will investigate this duality using the flow of a fast diffusion equation which has been considered in [START_REF] Del Pino | On the extinction profile for solutions of ut = ∆u (N-2)/(N+2)[END_REF]. In dimension d = 2, Onofri's inequality plays the role of Sobolev's inequality and is the dual of the logarithmic Hardy-Littlewood-Sobolev inequality according to [START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF][START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF][START_REF] Calvez | The parabolic-parabolic Keller-Segel model in R 2[END_REF]. Based on [START_REF]On a singular diffusion equation[END_REF][START_REF] Vázquez | The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane[END_REF], we will investigate this duality using a variant of the logarithmic diffusion equation, also known as the super-fast diffusion equation.

In a recent paper, [START_REF] Carlen | Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF], E. Carlen, J.A. Carrillo and M. Loss noticed that Hardy-Littlewood-Sobolev inequalities in dimension d ≥ 3 (or the logarithmic Hardy-Littlewood-Sobolev inequality if d = 2) and some special Gagliardo-Nirenberg inequalities can be related through another fast diffusion equation. Understanding the differences between the two approaches is one of the motivations of this paper.

Consider Sobolev's inequality in R d , d ≥ 3, (1) u 2 L 2 * (R d ) ≤ S d ∇u 2 L 2 (R d ) ∀ u ∈ D 1,2 (R d )
, where S d is the Aubin-Talenti constant (see [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF]) and 2 * = 2 d d-2 . The space D 1,2 (R d ) is defined as the completion of smooth solutions with compact support w.r.t. the norm w → w := ( ∇w 2 L 2 (R d ) + w 2 L 2 * (R d ) ) 1/2 . The Hardy-Littlewood-Sobolev inequality

(2)

S d v 2 L 2 d d+2 (R d ) ≥ R d v (-∆) -1 v dx ∀ v ∈ L 2 d d+2 (R d )
involves the same optimal constant, S d .

As it has been noticed in [START_REF] Lieb | Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities[END_REF], (1) and ( 2) are dual of each other, in the following sense. To a convex functional F , we may associate the functional F * defined by Legendre's duality as

(3) F * [v] := sup R d u v dx -F [u] .
For instance, to F 1 [u] = 1 2 u 2 L p (R d ) defined on L p (R d ), we henceforth associate

F * 1 [v] = 1 2 v 2 L q (R d ) on L q (R d )
where p and q are Hölder conjugate exponents: 1/p + 1/q = 1. The supremum can be taken for instance on all functions in L p (R d ), or, by density, on the smaller space of the functions u ∈ L p (R d ) such that ∇u ∈ L 2 (R d ). Similarly, to

F 2 [u] = 1 2 S d ∇u 2 L 2 (R d ) , we associate F * 2 [v] = 1 2 S -1 d R d v (-∆) -1 v dx where (-∆) -1 v = G d * v with G d (x) = 1 d-2 |S d-1 | -1 |x| 2-d .
As a straightforward consequence of Legendre's duality, if we have a functional inequality of the form

F 1 [u] ≤ F 2 [u], then we have the dual inequality F * 1 [v] ≥ F * 2 [v].
In this paper, we investigate the duality of ( 1) and (2) using a nonlinear diffusion equation. If v is a positive solution of the following fast diffusion equation:

(4) ∂v ∂t = ∆v m t > 0 , x ∈ R d ,
and if we define

H(t) := H d [v(t, •)], with H d [v] := R d v (-∆) -1 v dx -S d v 2 L 2 d d+2 (R d )
, then we observe that

1 2 H ′ = - R d v m+1 dx + S d R d v 2 d d+2 dx 2 d R d ∇v m • ∇v d-2 d+2 dx where v = v(t,
•) is a solution of (4). With the choice m = d-2 d+2 , we find that m + 1 = 2 d d+2 , so that the above identity can be rewritten with u = v m as follows.

Proposition 1.1. Assume that d ≥ 3 and m = d-2 d+2 . If v is a solution of (4) with nonnegative initial datum in L 2d/(d+2) (R d ), then 1 2 d dt R d v (-∆) -1 v dx -S d v 2 L 2 d d+2 (R d ) = R d v m+1 dx 2 d S d ∇u 2 L 2 (R d ) -u 2 L 2 * (R d ) ≥ 0 .
As a consequence, one can prove that (2), which amounts to H ≤ 0, is a consequence of (1), that is H ′ ≥ 0, by showing that lim sup t>0 H(t) = 0. In this way, we also recover the property that u = v m is an optimal function for (1) if v is optimal for [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF]. By integrating along the flow defined by (4), we can actually obtain optimal integral remainder terms which improve on the usual Sobolev inequality (1), but only when d ≥ 5 for integrability reasons: see Theorem 2.3. A slightly weaker result is the following improved Sobolev inequality, which relates (1) and (2). Theorem 1.2. Assume that d ≥ 5 and let q = d+2 d-2 . There exists a positive constant C such that, for any w ∈ D 1,2 (R d ), we have

S d w q 2 L 2 d d+2 (R d ) - R d w q (-∆) -1 w q dx ≤ C w 8 d-2 L 2 * (R d ) ∇w 2 L 2 (R d ) -S d w 2 L 2 * (R d ) .
Moreover, we know that

C ≤ 1 + 2 d 1 -e -d/2 S d .
Considerable efforts have been devoted to obtain improvements of Sobolev's inequality starting with [START_REF] Brezis | Sobolev inequalities with remainder terms[END_REF][START_REF] Bianchi | A note on the Sobolev inequality[END_REF]. On the whole euclidean space, nice results based on rearrangements have been obtained in [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] and we refer to [START_REF] Cianchi | Quantitative Sobolev and Hardy inequalities, and related symmetrization principles[END_REF] for an interesting review of the various improvements that have been established over the years. It has to be noted that they are all of different nature than the inequality in Theorem 1.2, which is the main result of this paper.

Our approach is based on the evolution equation ( 4). Many papers have been devoted to the study of the asymptotic behaviour of the solutions, in bounded domains: [START_REF] Berryman | Stability of the separable solution for fast diffusion[END_REF][START_REF] Galaktionov | Fast diffusion equation with critical Sobolev exponent in a ball[END_REF][START_REF] Savaré | The asymptotic profile of solutions of a class of doubly nonlinear equations[END_REF], or in the whole space: [START_REF] King | Self-similar behaviour for the equation of fast nonlinear diffusion[END_REF][START_REF] Peletier | Self-similar solutions of a fast diffusion equation that do not conserve mass[END_REF][START_REF] Galaktionov | Asymptotic behaviour near finite-time extinction for the fast diffusion equation[END_REF]. In particular, the Cauchy-Schwarz inequality has been repeatedly used, for instance in [START_REF] Berryman | Stability of the separable solution for fast diffusion[END_REF][START_REF] Savaré | The asymptotic profile of solutions of a class of doubly nonlinear equations[END_REF], and turns out to be a key tool in the proof of Theorem 1.2, as well as the solution with separation of variables, which is related to the Aubin-Talenti optimal function for (1). The novelty in our approach is to consider the problem from the point of view of the functional associated to (2) using( 4) with m = (d -2)/(d + 2).

We now turn our attention to the case of the dimension d = 2. As we shall see in Section 3, Onofri's inequality [START_REF] Onofri | On the positivity of the effective action in a theory of random surfaces[END_REF] (5) log

R 2 e g dµ - R 2 g dµ ≤ 1 16 π R 2 |∇g| 2 dx ∀ g ∈ D(R d )
plays the role of Sobolev's inequality in higher dimensions. Here the probability measure dµ is defined by

dµ(x) := µ(x) dx with µ(x) := 1 π (1 + |x| 2 ) 2 ∀ x ∈ R 2 .
As for Sobolev's inequality, duality can also be used. The dual of Onofri's inequality is the logarithmic Hardy-Littlewood-Sobolev inequality: for any

f ∈ L 1 + (R 2 ) with M = R 2 f dx, such that f log f , (1 + log |x| 2 ) f ∈ L 1 (R 2 ), we have (6) R 2 f log f M dx + 2 M R 2 ×R 2 f (x) f (y) log |x -y| dx dy + M (1 + log π) ≥ 0 .
The duality has been established on the two-dimensional sphere in [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF][START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF] and directly on the euclidean space R 2 in [START_REF] Calvez | The parabolic-parabolic Keller-Segel model in R 2[END_REF]. The euclidean case can also be recovered from the inequality on the sphere using the stereographic projection. For completeness, we shall give a proof of the inequality (with optimal constants) in the R 2 case: see Proposition 3.2. The logarithmic Hardy-Littlewood-Sobolev inequality has recently attracted lots of attention in connection with the Keller-Segel model or in geometry: see for instance [START_REF] Calvez | The parabolic-parabolic Keller-Segel model in R 2[END_REF][START_REF] Dolbeault | Optimal critical mass in the two-dimensional Keller-Segel model in R 2[END_REF][START_REF] Okikiolu | Extremals for logarithmic Hardy-Littlewood-Sobolev inequalities on compact manifolds[END_REF].

We may now proceed in the case d = 2 as we did for d ≥ 3. Let

H 2 [v] := R 2 (v -µ) (-∆) -1 (v -µ) dx - 1 4 π R 2 v log v µ dx .
Assume that v is a positive solution of ( 7)

∂v ∂t = ∆ log v µ t > 0 , x ∈ R 2 ,
which replaces (4). Then we have the analog of Proposition 1.1.

Proposition 1.3. Assume that d = 2. If v is a solution of (7) with nonnegative initial datum v 0 in L 1 (R 2 ) such that R 2 v 0 dx = 1, v 0 log v 0 ∈ L 1 (R 2 ) and v 0 log µ ∈ L 1 (R 2 ), then d dt H 2 [v(t, •)] = 1 16 π R 2 |∇u| 2 dx - R 2 e u 2 -1 u dµ with log(v/µ) = u/2.
The right hand side is nonnegative by Onofri's inequality:

d dt H 2 [v(t, •)] ≥ 1 16 π R 2 |∇u| 2 dx + R 2 u dµ -log R 2
e u dµ ≥ 0 .

See Lemma 3.3 and Corollary 3.4 for details, and Section 3.4 for further considerations on the two-dimensional case.

Improved Sobolev inequalities

This section is devoted to the proof of Theorem 1.2. We shall assume that

m = d -2 d + 2 and d ≥ 3 .
From the computations of Section 1, it is clear that the maximum of H d [v] is achieved if and only if u = v m is an extremal for Sobolev's inequality. This is of course consistent with the fact that extremal points for Hardy-Littlewood-Sobolev inequalities and Sobolev inequalities are related through Legendre's duality precisely by the relation

u = v m .
It is also straightforward to check that (4) admits special solutions with separation of variables such that, for any T > 0, 2) . Such a solution vanishes at t = T and this behaviour is generic in a large class of solutions. Define [START_REF] Del Pino | On the extinction profile for solutions of ut = ∆u (N-2)/(N+2)[END_REF][START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] For any solution v of (4) with nonnegative, continuous, not identically zero initial datum v 0 ∈ L 2d/(d+2) (R d ), there exists T > 0, λ > 0 and x 0 ∈ R d such that v(t, •) ≡ 0 for any t ∈ (0, T ) and

v T (t, x) = c (T -t) α (F (x)) d+2 d-2 ∀ (t, x) ∈ (0, T ) × R d , where α = (d + 2)/4, c 1-m = 4 m d, m = d-2 d+2 , p = d/(d -2) and F is the Aubin- Talenti solution of -∆F = d (d -2) F (d+2)/(d-
v * := sup x∈R d (1 + |x| 2 ) d+2 |v(x)| . Lemma 2.1.
lim t→T- (T -t) -1 1-m v(t, •)/v(t, •) -1 * = 0 with v(t, x) = λ (d+2)/2 v T (t, (x -x 0 )/λ).
We shall refer to such a solution as a solution vanishing at time T . The above result has been established first in [START_REF] Del Pino | On the extinction profile for solutions of ut = ∆u (N-2)/(N+2)[END_REF] when v 0 * is finite and extended to solutions corresponding to any initial data [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]Theorem 7.10]. In this framework, it is easy to establish further a priori estimates as follows. (R d ), then for any t ∈ (0, T ) we have the estimates

v 0 ∈ L 2d/(d+2) + (R d ) in
4 (T -t) (d+2) S d d 2 ≤ R d v m+1 (t, x) dx ≤ R d v m+1 0 dx , ∇v m (t, •) 2 L 2 (R d ) ≥ S -1 d 4 (T -t) d+2 d 2 -1
, and the vanishing time T is bounded by

T ≤ 1 4 (d + 2) S d R d v m+1 0 dx 2 d . If additionnally d ≥ 5, then T ≥ d+2 2 d R d v m+1 0 dx ∇v m 0 -2 L 2 (R d ) and R d v m+1 (t, x) dx ≥ R d v m+1 0 dx -2 d d+2 t ∇v m 0 2 L 2 (R d ) , ∇v m (t, •) 2 L 2 (R d ) ≤ ∇v m 0 2 L 2 (R d ) .
Proof. By definition of the vanishing time, we find that J(t) := R d v(t, x) m+1 dx satisfies J(t) > 0 for any t ∈ (0, T ), J(T ) = 0 and, using the equation and integrating by parts,

J ′ = -(m + 1) ∇v m 2 L 2 (R d ) ≤ - m + 1 S d J 1-2 d .
If d ≥ 5, then we also have

J ′′ = 2 m (m + 1) R d v m-1 (∆v m ) 2 dx ≥ 0 .
It is easy to check that such an estimate makes sense if v = v T . For a general solution, this is also true as can be seen by rewriting the problem on S d as in [START_REF] Del Pino | On the extinction profile for solutions of ut = ∆u (N-2)/(N+2)[END_REF]. It is then clear that integrability conditions for v are exactly the same as for v T . By integrating the first inequality from t to T , we find that -J(t)

2 d = J(T ) 2 d -J(t) 2 d ≤ -4 (T -t) (d+2) S d ∀ t ∈ [0, T ) ,
which gives an upper bound for T by choosing t = 0. The second inequality shows the decay of t → ∇v m (t, •) 2 L 2 (R d ) and gives the estimate 0 = J(T ) ≥ J(0) + T J ′ (0), thus providing a lower bound for T .

For later purpose, let us notice that (8)

J ′ J ≤ - m + 1 S d J -2 d ≤ -κ with κ := 2 d d + 2 1 S d R d v m+1 0 dx -2 d ≤ d 2 T .
Next, we are going to compute the second derivative of

H(t) = H d [v(t,
•)] w.r.t. t along the flow of (4). For this purpose, we assume that d ≥ 5 and notice that by the Cauchy-Schwarz inequality, we have

∇v m 4 L 2 (R d ) = R d v (m-1)/2 ∆v m • v (m+1)/2 dx 2 ≤ R d v m-1 (∆v m ) 2 dx R d v m+1 dx .
As a consequence, we get that

Q(t) := ∇v m (t, •) 2 L 2 (R d ) R d v m+1 (t, x) dx -(d-2)/d is monotone decreasing. More precisely, with m = (d -2)/(d + 2), (9) Λ(t) := R d |∇(v(t, x)) m | 2 dx R d (v(t, x)) m+1 dx and G(t 1 , t 2 ) := exp (m + 1) t2 t1 Λ(s) ds , we get that Q ′ = -2 m J 2 d -1 K with K := R d v m-1 |∆v m + Λ v| 2 dx and, from H ′ = 2 J (S d Q -1) ,
we deduce that (10)

H ′′ = J ′ J H ′ + 2 J S d Q ′ = -(m + 1) Λ H ′ -4 m S d J 2 d K .
As a consequence, the standard Sobolev inequality can be improved by an integral remainder term.

Theorem 2.3. Assume that d ≥ 5 and let q = d+2 d-2 . For any w ∈ D 1,2 (R d ) such that w q * < ∞, we have

S d w q 2 L 2 d d+2 (R d ) - R d w q (-∆) -1 w q dx + 4 q S d T 0 dt t 0 J 2 d (s) K(s) G(t, s) ds = 2 w 4 d-2 L 2 * (R d ) ∇w 2 L 2 (R d ) -S d w 2 L 2 * (R d ) T 0 G(t, 0) dt .
Here v is the solution of (4) with v 0 = w q , m = (d -2)/(d + 2) = 1/q, T is the vanishing time, Λ and G are defined by [START_REF] Cianchi | Quantitative Sobolev and Hardy inequalities, and related symmetrization principles[END_REF], and we recall that

J(t) = R d v m+1 (t, x) dx, K(t) = R d v m-1 (t, x) |∆v m (t, x) + Λ(t) v(t, x)| 2 dx.
Proof. The identity follows from [START_REF] Cianchi | The sharp Sobolev inequality in quantitative form[END_REF] after an integration from 0 to t and another one from 0 to T . Details are left to the reader.

Recall that the standard Hardy-Littlewood-Sobolev inequality (2) amounts to

S d w q 2 L 2 d d+2 (R d ) - R d w q (-∆) -1 w q dx = -H d [w q ] ≥ 0 .
The main drawback of Theorem 2.3 is that several quantities can be computed only through the evolution equation and are therefore non explicit. With simple estimates, we can however get rid of such quantities. This is the purpose of Theorem 1.2.

Proof of Theorem 1.2. Notice that

w q 2 L 2d/(d+2) (R d ) = w 1+2/d L 2 * (R d )
, so that the inequality holds in the space D 1,2 (R d ). Theorem 1.2 can be established first in the setting of smooth functions such that w q * < ∞ and then arguing by density.

From ( 8) and ( 10), we know that

H ′′ ≤ -κ H ′ with κ = 2 d d + 2 1 S d R d v m+1 0 dx -2/d
. By writing that -H(0) = H(T ) -H(0) ≤ H ′ (0) (1e -κ T )/κ and using the estimate κ T ≤ d/2, we obtain the result.

3. The two-dimensional case 3.1. From Gagliardo-Nirenberg inequalities to Onofri's inequality. Consider the following sub-family of Gagliardo-Nirenberg inequalities [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF] f

L 2p (R d ) ≤ C p,d ∇f θ L 2 (R d ) f 1-θ L p+1 (R d ) with θ = θ(p) := p-1 p d d+2-p (d-2) , 1 < p ≤ d d-2 if d ≥ 3 and 1 < p < ∞ if d = 2
. Such an inequality holds for any smooth function f with sufficient decay at infinity and, by density, for any function f ∈ L p+1 (R d ) such that ∇f is square integrable. We shall assume that C p,d is the best possible constant. In [START_REF] Del Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF], it has been established that equality holds in [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF]

if f = F p with (12) F p (x) = (1 + |x| 2 ) -1 p-1 ∀ x ∈ R d
and that all extremal functions are equal to F p up to a multiplication by a constant, a translation and a scaling. If d ≥ 3, the limit case p = d/(d -2) corresponds to Sobolev's inequality and one recovers the results of T. Aubin and G. Talenti in [START_REF] Aubin | Problèmes isopérimétriques et espaces de Sobolev[END_REF][START_REF] Talenti | Best constant in Sobolev inequality[END_REF], with θ = 1: the optimal functions for (1) are, up to scalings, translations and multiplications by a constant, all equal to

F d/(d-2) (x) = (1 + |x| 2 ) -(d-2)/2
, and

S d = (C d/(d-2), d ) 2 .
When p → 1, the inequality becomes an equality, so that we may differentiate both sides with respect to p and recover the euclidean logarithmic Sobolev inequality in optimal scale invariant form (see [START_REF] Gross | Logarithmic Sobolev inequalities[END_REF][START_REF] Weissler | Logarithmic Sobolev inequalities for the heat-diffusion semigroup[END_REF][START_REF] Del Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] for details). For completeness, let us mention that the fast diffusion equation ( 4) has deep connection with [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF] when p = 1/(2 m -1), but this is out of the scope of the present paper: see [START_REF] Del Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF] for more details on this issue. We now investigate the limit p → ∞ in (11) when d = 2: Onofri's inequality ( 5) can be obtained in the limit. As an endpoint case of the family of the Gagliardo-Nirenberg inequalities [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF], it plays the role of Sobolev's inequality in dimension d ≥ 3.

Proposition 3.1. Assume that g ∈ D(R d ) is such that R 2 g dµ = 0 and let f p := F p (1 + g 2p ),
where F p is defined by [START_REF]On a singular diffusion equation[END_REF]. Then we have

1 ≤ lim p→∞ C p,2 ∇f θ(p) L 2 (R 2 ) f 1-θ(p) L p+1 (R 2 ) f L 2p (R 2 ) = e 1 16 π R 2 |∇g| 2 dx R 2 e g dµ .
We recall that µ(x) := 1 π (1 + |x| 2 ) -2 , and dµ(x) := µ(x) dx. Proof. We can rewrite [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF] as

R 2 |f | 2p dx R 2 |F p | 2p dx ≤ R 2 |∇f | 2 dx R 2 |∇F p | 2 dx p-1 2 R 2 |f | p+1 dx R 2 |F p | p+1 dx
and observe that, with f = f p , we have:

(i) lim p→∞ R 2 |F p | 2p dx = R 2 1 (1+|x| 2 ) 2 dx = π and lim p→∞ R 2 |f p | 2p dx = R 2 F 2p p (1 + g 2p ) 2p dx = R 2 e g (1 + |x| 2 ) 2 dx so that the R 2 |f | 2p dx/ R 2 |F p | 2p dx converges to R 2 e g dµ as p → ∞, (ii) R 2 |F p | p+1 dx = (p -1) π/2, lim p→∞ R 2 |f p | p+1 dx = ∞, but lim p→∞ R 2 |f p | p+1 dx R 2 |F p | p+1 dx = 1 ,
(iii) expanding the square and integrating by parts, we find that

R 2 |∇f p | 2 dx = 1 4p 2 R 2 F 2 p |∇g| 2 dx - R 2 (1 + g 2p ) 2 F p ∆F p dx = 1 4p 2 R 2 |∇g| 2 dx + 2π p + 1 + o(p -2
) .

Here we have used R 2 |∇F p | 2 dx = 2π p+1 and the condition R 2 g dµ = 0 in order to discard one additional term of the order of p -2 . On the other hand, we find that

R 2 |∇f | 2 dx R 2 |∇F p | 2 dx p-1 2 ∼ 1 + p + 1 8π p 2 R 2 |∇g| 2 dx p-1 2 ∼ e 1 16 π R 2 |∇g| 2 dx
as p → ∞. Collecting these estimates concludes the proof.

v ∈ L 1 + (R 2 ) with R 2 v dx = 1, such that v log v and (1 + log |x| 2 ) v ∈ L 1 (R 2 ), we have F * 1 [v] -F * 2 [v] = R 2 v log v µ dx -4 π R 2 (v -µ) (-∆) -1 (v -µ) dx ≥ 0 .
This result has been observed by several authors: see for instance [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF][START_REF] Carlen | Competing symmetries, the logarithmic HLS inequality and Onofri's inequality on S n[END_REF] for the duality argument on the two-dimensional sphere, which gives the above inequality on R 2 by the stereographic projection. For completeness, let us give a proof. We essentially follow the computation of [START_REF] Calvez | The parabolic-parabolic Keller-Segel model in R 2[END_REF]Appendix] but explicitly compute the constants.

Proof. Taking the infimum in (3) on the Orlicz space

u ∈ L 1 loc (R 2 ) : ∇u ∈ L 2 (R 2 ) , R 2 e u dµ < ∞ ,
we find that

F * 1 [v] = R 2 u v dx -log R 2 e u dµ with log v = log µ + u -log R 2 e u dµ , F * 2 [v] = R 2 u (v -µ) dx - 1 16 π R 2 |∇u| 2 dx with -∆u = 8 π (v -µ) ,
which proves the result.

A useful observation is the fact that -∆ log µ = 8 π µ can be inverted as

(-∆) -1 µ = 1 8 π log µ + C .
It is then easy to check that u := (-∆)

-1 µ = G 2 * µ, with G 2 (x) = -1 2π log |x|, is such that u(0) = 0, which determines C = 1 8 π log π.
Hence with the notations of Proposition 3.2, we may observe that

F * 1 [v] = R 2 v log v µ dx , F * 2 [v] = 4 π R 2 v (-∆) -1 v dx - R 2 v log µ dx -1 -log π .
Collecting these observations, we find that

F * 1 [v] -F * 2 [v] = R 2 v log v dx -4 π R 2 v (-∆) -1 v dx + 1 + log π ≥ 0 . For any f ∈ L 1 + (R 2 ) with M = R 2 f dx, such that f log f , (1 + log |x| 2 ) f ∈ L 1 (R 2
), this inequality, written for v = f /M , is nothing else than the logarithmic Hardy-Littlewood-Sobolev inequality (5).

Logarithmic diffusion equation: Onofri and logarithmic Hardy

-Little- wood-Sobolev inequalities. The computation of d dt H 2 [v(t,
•)] Section 1 is formal but can easily be justified after noticing that the image w of v by the inverse stereographic projection on the sphere S 2 , up to a scaling, solves the equation ∂w ∂t = ∆ S 2 w .

More precisely, if x = (x 1 , x 2 ) ∈ R 2 , then u and w are related by

w(t, y) = u(t, x) 4 π µ(x) , y = 2 (x1,x2) 1+|x| 2 , 1-|x| 2 1+|x| 2 ∈ S 2 .
See [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]Section 8.2] for a review of some known results for the logarithmic diffusion equation, or super-fast diffusion equation, ∂v ∂t = ∆ log v, and [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF][START_REF] Vázquez | The fast diffusion equation with logarithmic nonlinearity and the evolution of conformal metrics in the plane[END_REF] for earlier results. Necessary adaptations to the case of ( 7) are left to the reader. Striking properties are the facts that the solution of ( 7) globally exists and its mass is preserved. For simplicity, we shall therefore assume that 1 = R 2 v 0 dx = R 2 v(t, x) dx for any t ≥ 0 and recall that R 2 µ dx = 1. It turns out that d dt H 2 [v(t, •)] has a sign because of Onofri's inequality.

Lemma 3.3. For any u ∈ D(R d ) such that R 2 e u 2 dµ = 1, we have 1 16 π R 2 |∇u| 2 dx ≥ R 2 e u 2 -1 u dµ .
This inequality can of course be extended by density to the natural Orlicz space for which all integrals are well defined.

Proof. Inspired by [START_REF] Lata | Between Sobolev and Poincaré[END_REF], let us define Let us prove that h ′′′ (t) > 0 for t ∈ (0, 1) a.e. If R 2 u e tu dµ > 0, define

h(t) := log R 2 e tu dµ . Claim 1: h is convex and h ′ is convex. Let us observe that R 2 e tu dµ 2 h ′′ (t) = R 2 e tu dµ R 2 u 2 e tu dµ -
dν t := e tu R 2 e tu dµ dµ and v := u R 2 u dν t , so that R 2 v dν t = 1. Then h ′′′ (t) has the sign of R 2 (v 3 -3 v 2 + 2) dν t .
Taking the constraint into account, a direct optimization shows that an optimal function v takes two values, 1 ± a for some a > 0 and moreover, ν t ({v = 1 + a}) = ν t ({v = 1 -a}) because of the condition R 2 v dν t = 1. In such a case, it is straightforward to check that R 2 (v 3 -3 v 2 + 2) dν t = 0, thus proving that h ′′′ (t) ≥ 0. If R 2 u e tu dµ < 0, a similar computation with v := -u/ R 2 u dν t shows that -h ′′′ (t) has the sign of R 2 (-v 3 -3 v 2 + 2) dν t under the condition R 2 v dν t = -1, thus proving again that h ′′′ (t) ≥ 0.

Since h ′ (t) = R 2 u dν t is monotone increasing, R 2 u e tu dµ = 0 occurs for at most one t ∈ [0, 1] and we can conclude that h ′ is convex.

Claim 2: h(1) ≥ h ′ (1/2). We know that h(0) = h(1/2) = 0 and, by convexity of h ′ , h ′′ is monotone nondecreasing. Let κ := h ′′ (1/2) and p := h ′ (1/2). On the one hand, we have h ′′ (t) ≤ κ for any t ∈ [0, 1/2], which means that, after integrating from t < 1/2 to 1/2, we have 1 2 ] . Hence, one more integration from 0 to 1/2 gives

p -h ′ (t) ≤ κ ( 1 2 -t) ∀ t ∈ [0,
p 2 = [p t -h(t)] 1/2 0 ≤ κ t 2 (1 -t) 1/2 0 = κ 8 .
On the other hand, we have h ′′ (t) ≥ κ for any t ∈ [1/2, 1], which means that, after integrating from 1/2 to t > 1/2, we have

h ′ (t) -p ≥ κ (t -1 2 ) ∀ t ∈ [ 1 2 , 1] .
One more integration from 1/2 to 1 gives

h(1) - p 2 ≥ κ t 2 (t -1) 1/2 0 = κ 8 .
Collecting the two estimates, we find that h(1) ≥ p, which proves the claim.

Using h(0) = log R 2 dµ = 0 and h(1/2) = log R 2 e u/2 dµ = 0, we have found that p = h ′ (1/2) = R 2 u e u/2 dµ ≤ h(1) = log R 2 e u dµ . This proves that

d dt H 2 [v(t, •)] ≥ 1 16 π R 2 |∇u| 2 dx + R 2 u dµ -log R 2 e u dµ ≥ 0 ,
where the last inequality is nothing else than Onofri's inequality.

Corollary 3.4. Consider a solution of (7) a nonnegative initial datum v 0 such that

R 2 v 0 dx = 1. Then d dt H 2 [v(t, •)] ≥ 0. 3.4.
Towards an improved Onofri inequality ? At least at a formal level, we can observe that computations of Section 2 for d ≥ 3 do not carry to the case d = 2. With u = 2 log(v/µ), Equation ( 7) becomes

µ ∂u ∂t = e -u 2 ∆u , so that J := 2 R 2 v (log (v/µ) -1) dx takes the form J = R 2 e u 2 (u -2) dµ
and, when computing along the flow, we get

J ′ = - 1 2 R 2 |∇u| 2 dx and J ′′ = R 2 (∆u) 2 e -u 2 dx µ .
If we consider the Cauchy-Schwarz inequality as in Section 2, namely

4 J ′ 2 = R 2 |∇u| 2 dx 2 = R 2 ∆u u dx 2 = R 2 ∆u e -u/4 1 √ µ • u e u/4 √ µ dx 2 ≤ J ′′ R 2 u 2 e u 2 dµ ,
it clearly turns out that the above r.h.s. cannot be controlled by J J ′′ using a simple Cauchy-Schwarz inequality. The scheme of the proof of Theorem 1.2 for improving Sobolev's inequality cannot be directly applied to Onofri's inequality.

Concluding remarks

In [START_REF] Carlen | Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF], E. Carlen, J.A. Carrillo and M. Loss noticed that Hardy-Littlewood-Sobolev inequalities and Gagliardo-Nirenberg inequalities can be related through the fast diffusion equation ( 4) with exponent m = d/(d + 2), when d ≥ 3. The key computation goes as follows: •)] has a sign because of the Gagliardo-Nirenberg inequalities [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF]. Using the fact that the asymptotic behaviour of the solutions of the fast diffusion equation ( 4) with m = d/(d + 2) is governed by the Barenblatt self-similar solutions (see [START_REF] Del Pino | Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions[END_REF]), an integral remainder term (obtained by integrating along the flow of ( 4)) has been established, which improves on Hardy-Littlewood-Sobolev inequalities [START_REF] Beckner | Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality[END_REF].

1 2 d dt H d [v(t, •)] = 1 2 d dt R d v (-∆) -1 v dx -S d v 2
A similar result holds for the logarithmic Hardy-Littlewood-Sobolev inequality in dimension d = 2. Computing along the flow of (4) with exponent m = 1/2, it turns out that

v L 1 (R 2 ) 8 d dt 4 π v L 1 (R 2 ) R 2 v (-∆) -1 v dx - R 2 v log v dx = u 4 L 4 (R 2 ) ∇u 2 L 2 (R 2 ) -π v 6 L 6 (R 2 )
, which is again one of the Gagliardo-Nirenberg inequalities [START_REF] Daskalopoulos | On fast diffusion nonlinear heat equations and a related singular elliptic problem[END_REF], the one corresponding to d = 2 and q = 3, which is moreover such that π (C 3, 2 ) 6 = 1.

The results of [START_REF] Carlen | Loss, Hardy-Littlewood-Sobolev inequalities via fast diffusion flows[END_REF] relate Hardy-Littlewood-Sobolev inequalities (the logarithmic Hardy-Littlewood-Sobolev inequality if d = 2) with non-critical Gagliardo-Nirenberg inequalities through an evolution equation, although the Barenblatt self-similar solutions associated to the diffusion equation are not optimal for such Gagliardo-Nirenberg inequalities. In such a setting as well as in the setting considered in Sections 1-3, a nonlinear flow allows to improve on well known inequalities, with rather straightforward computations. These two examples suggest that much more can be done using flows of nonlinear diffusion equations.

Lemma 2 . 2 .

 22 Let d ≥ 3 and m = (d -2)/(d + 2). If v is a solution of (4) vanishing at time T > 0 with initial datum v 0 ∈ L 2d/(d+2) +

3. 2 . 2 e u dµ and F 2 [Proposition 3 . 2 .

 22232 Legendre duality. Now we study the duality which relates Onofri's inequality (5) and the logarithmic Hardy-Littlewood-Sobolev inequality (6) in R 2 . With F 1 [u] := log R inequality amounts to F 1 [u] ≤ F 2 [u]. As for the case d ≥ 3, we use Legendre's transformation (3). For any

R 2 u e tu dµ 2 ≥ 2 e tu dµ 3 h 2 u 3 e tu dµ R 2 e tu dµ 2 - 3 R 2 u 2 e tu dµ R 2 u e tu dµ R 2 e tu dµ + 2 R 2 u e tu dµ 3 .

 22232223222223 0 by the Cauchy-Schwarz inequality. With one more derivation, we find that R ′′′ (t) = R

L

  q+1 (R d ) ∇u 2 L 2 (R d )u 2q L 2q (R d )with u = v (d-1)/(d+2) and q = (d + 1)/(d -1). An explicit computation shows thatd (d-2) (d-1) 2 S d = (C q, d ) 2q , thus proving that d dt H d [v(t,

J. Dolbeault: Ceremade, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France.E-mail address: dolbeaul@ceremade.dauphine.fr

Acknowlegments. The author thanks J.A. Carrillo and M.J. Esteban for useful suggestions. This work has been partially supported by the projects CBDif and EVOL of the French National Research Agency (ANR). c 2010 by the author. This paper may be reproduced, in its entirety, for non-commercial purposes.


Dr Hector G Lopez-Ruiz
email: hlopezruiz@gmail.com

Planning for resilience in urban transport

Keywords: Modelling, Greenhouse gas, Coupling, Scenarios, 2050 horizon, mitigation, adaptation, multimodality, decoupling

This paper develops the idea of taking into account the systemic effects of public policies for sustainable transport in long-term scenarios. A comprehensive assessment of these effects is a particularly important factor in identifying potential unbalancing elements and planning for resilience to the effects of mitigation policies in view of increasing acceptance to different measures on which GHG reductions depend.

I. INTRODUCTION

In an effort to counteract global warming and lessen the geopolitical risks linked to energy supplies by decreasing oil intensity in economic activities, many countries have decided to limit their oil consumption and global greenhouse (GHG) emissions. This determination is requiring governments to make a choice between -the well-known-oil dependent growth path and the -rather unfamiliar-green growth path.

As with any economic change concerning energy supplies, uneasiness about cutting GHG emissions stem -in part-from the uncertainty concerning technological options for producing enough "zero-emission" energy in order to cover the world's demand. What is more, the prospect of having to change the way societies are organized in order to cut down on carbonintensive behaviours is also causing unrest. Even though growing constraints on fossil-fuels are already curbing energy demand and modifying behaviour in some activities, nervousness around the fact that not all sectors of the economy are equally flexible and adaptive has surfaced.

In the transport sector, the undeniable suppleness and low costs of road transportation (for passengers and freight) has been a key element to its success over the last few decades and largely explains its increasing demand. This growth trend in mobility has been particularly vigorous in urban transport and has become an important issue in the debate on developing sustainable mobility.

The question of rising urban mobility and its inherent urban sprawl is often linked to, both, economic growth [START_REF] Echenique | Mobility and Income[END_REF] and high levels of GHG emissions. The idea of having to choose between growth and the environment has been one of the main debates in planning for the reduction of the climate change risk linked to human activities.

Recent scientific insight shows that in order to decrease the probability of overshooting a 2°C increase in global temperatures, GHG emissions should be cut by at least 50 % (from the 1990 level) in the next 40 years. Consequently, certain studies have implied that cutting down drastically on road transportation in industrialized countries and curbing road transport demand growth in newly industrialized countries will, inevitably, translate into an economic downturn.

In an effort to offer new and innovative solutions aiming at reducing emissions, numerous studies have analysed different courses of action and their implications on the transport sector. All of them conclude that although GHG mitigation will not be an easy task, it does not necessarily suppose shrinking economic activity. have looked into several options on how to plan for these drastic reductions. These studies concur on the fact that new technologies and their widespread use will be necessary in order to attain considerable GHG reductions. Furthermore, they also agree that these new technologies will not be enough for countries to get to their GHG objectives. Indeed, most works conclude that GHG mitigation supposes careful planning in order to increase the match between "green" transport supply and consumer demand through the use of incentive economic instruments. Thus, they clearly underline that expected technological progress in the transport sector cannot be effective if it is not accompanied by deep organizational and behavioural changes.

Consequently, a growing body of research has developed interest on how public policies aiming at GHG mitigation will have an impact on socio-ecological systems. Along these lines, this paper will explore how a simple microeconomic model can help plan for resilience of the transport system in its quest for sustainability through a mix of behaviour change and new technologies.

In order to carry out our analysis we will refer to what is commonly accepted as the two main options for GHG mitigation through changes in behaviour: increasing the use of other transport modes (multimodality) or moving less (decoupling transport from GDP growth).

Both of these options suppose changes in how everyday mobility is carried out as well as important modifications in how people organize their lives. From an economic standpoint, the two main factors in which behaviour oriented policies have an impact are: time budgets and money budgets. Many authors have proposed different models referring to the relationship existing between the two, but, for the needs of this paper, we will be referring particularly to the LINDER (1970) model. This approach will enable us to give new insight on how planning for behavioural change in transport activities through time-intensiveness management can offer better adaptive capacity of the system to environmental constraints.

For this, we will propose a simple framework based on a time-value management approach that aims at exploring how policy analysis can be used to define long-term strategies based on time intensiveness. The main results will look into how both industrialized and newly industrialized economies face similar dilemmas when it comes to planning for long-term sustainability in transport activities.

II. THE ANALYTICAL FRAMEWORK

As planning for GHG mitigation goes forward, different studies have shown that, in the absence of new -revolutionary-motor technology, public policies aiming at deep behavioural changes will be necessary in order to curb GHG emissions. These changes imply increasing constraints on high carbon footprint mobility through economic and social incentives that could range from educational and awareness programs to tolling, rationing or even carbon taxing.

Because of their socio-economic characteristics, the wide array of public policies that can be carried out in order to curb GHG emissions are tightly linked to changes in current mobility practices. Consequently, although these changes in behaviour represent a viable solution for mitigation of GHG emissions, they could also lead to imbalances on a microeconomic level. What is more, these imbalances could turn into counterproductive socio-economic side effects that could stun overall acceptance of the policy's objectives. As an answer to these preoccupations, an increasing body of work in social-ecological systems suggests the idea of planning for resilience [START_REF] Folke | Resilience: The emergence of a perspective for socialecological systems analyses[END_REF] in order to increase a system's adaptability and in consequence, policy acceptance. This planning process implies understanding that the implementation of different long-term policy mixes entails a (re-)optimization of agents' (passengers and firms) choices based on land-use and transport characteristics of a given territory. In other words, planning for resilience supposes a careful analysis of what different sustainable scenarios imply on welfare variations associated to trade-offs between transport expenditure; house/firm location; accessibility; transport money/time budgets and in fine consumption of goods/services. Consequently, in order to define how acceptance of different public policies can be increased through planning we need to asses the type of (re)-optimizations at play. For this, we propose a simple microeconomic choice model that lets us understand how, according to past tendencies (characterized by the coupling between growth and mobility), future public policies will impact demand for transport services as well as trade-offs linked to behavioural change and infrastructure use on different geographical scales. This model relies on the idea of a representative agent that optimizes her/his transport decisions by taking into account opportunity (defined as the sum of goods and services that can be consumed in a period of time, LINDER, 1970) and cost in respect to mode characteristics and infrastructure level of service (measured through a lateness index).

The model considers that the lateness index is defined by the difference existing between normal transit time and real transit time. This indicator is useful in factoring in speed, distance and time into the calculation of the choice model and has the convenience of being comparable between modes.

In this manner, the proposed framework allows us to assess the representative agent's choices that are coherent with the transport structure and its level of service. In the model, the value assigned to each choice is calculated using the following equations:

        i Ni N j t t t a              ) ( ) ( ) ( 1

Equation 1

Where:

t time acces time ies opportunit cos * ) (    Equation 2 and ) ( ) ( t index Lateness t   Equation 3
Using this microeconomic model, the inherent logic for technico-organizational public policy assessments is linked to the idea that: public policies are implemented as increasing/decreasing constraints on the overall system in view of getting to a certain objective (for example, a defined reduction in oil consumption or a certain level of GHG emissions). Consequently, this implies a wide variety of social effects on different levels and aspects that will modify the agent's decisions based on changes in transport prices, travel times, level of service, etc. These effects are translated into the model as changes in the opportunities and/or cost factors. Furthermore, the changes in the opportunity/cost structure are then linked to household budget (money and time) reallocation effects that will have an important impact on other sectors (which will also have a loopback effect on transport).

III. ADAPTING TO CHANGE

The inherent principle of most sustainable transport policies is based on: rendering high carbon footprint transport less attractive (through prices, modal speeds, infrastructure level of service, awareness courses, etc.) than low carbon footprint transport. This principle aims at modifying individual microeconomic behaviours in order to have an impact on the overall transport structure. Although these effects will be different for each country, network and set of public policies, certain basic general economic trade-offs are valid in most analysis.

In essence, rendering high carbon footprint transport less attractive suggests the need for constraints on emissions. These constraints come into play as a signal aiming at changing behaviour patterns in transport users. Consequently, the socio-economic system adapts through a sharp increase in the use of low carbon footprint modes.

As constraints on high emission modes kick in, potential unbalancing factors could come into play. Indeed, in the presented microeconomic model, policy aiming at developing strong multimodality in a system, translates into: paying more (especially for car users who would be unable to shift to public transport) for lower transport speeds (an all modes), thus higher transport times. This situation supposes a loss of potential value added time that could, otherwise, be spent increasing revenue.

In sum, a multimodal scenario is based on mixing: an increase in the cost of car use with lower speeds in public transport:

t time acces time cos )* ( ies opportunit     
. This translates into a situation where an agent's cost structure grows as his set of opportunities would be -at beststable. What is more, this situation could also represent a risk to poor households that could suffer a "choking effect" through this double logic of: increasing transport costs and diminishing opportunities (UBBELS & VERHOEF (2006)). Consequently, the question of the widespread acceptance of policies aiming to develop multimodality is raised.

If we follow the same line of reasoning, the logic behind a decoupling scenario is very influenced by proximity services. In this type of scenario, public policies at play are largely related to spatial planning and infrastructure investment. In a decoupling logic, the main trade-off at play is directly linked to location strategies and production organization aimed at decoupling transport distances from GDP growth. This entails a densification of main cities and production sites through the optimization of land use. In other words, the  ratio would increase costs but would leave access/transport times stable (or even decrease them)

t time acces time cos )* ( ies opportunit     .
In this manner, transport cost characteristics in a decoupling logic lead to more stable transport money and time budgets. This is explained by the rise in transport prices being followed by land use optimization. Nevertheless, if a decoupling scenario were not followed by an adaptation strategy based on a fast increase in the supply of proximity opportunities, an important loss of welfare could be observed. Indeed, what good is creating dense cities without putting opportunities inside them? Through this analysis of policy effects on the transport system using a choice model we aim to show that the need to ensure a system's adaptability to change should be strongly linked to strategies looking to counterbalance undesirable effects. In this manner, the idea of increasing opportunities linked to transport activities should become a part of the sustainability planning agenda.

This translates into the need for an optimization of land use based on PHELPS's and BAUMOL's view of the Linder theorem. In their 1973 papers, they comment their views on time and consider that money does not buy time but it can buy time-intensiveness and thus, more opportunities. Paradoxically, the LINDER-PHELPS-BAUMOL analysis has been used for a long time in order to explain how, over the decades, society has increased timeintensiveness through access to speedier transport.

Nevertheless the Internet has showed us that it is not only physical speed that counts but rather the speed in which we can access opportunities. This suggests that the key element in sustainable transport planning lies in increasing the sum of goods and services (opportunities) linked to the accessibility offered by transport. Through this, the equilibrium in the  ratio is ensured as well as the system's resilience to environmental constraints.

Moreover, the need to adapt to the effects of mitigation policies does not only relate to an unbalanced  factor. This need is reinforced by the fact that as constraints on oil consumption grow, more and more passengers will turn to public transport services. Consequently, a second factor that explains the need for planning for resilience is: a shift in market power.

As private vehicle costs rise, public transport use will also rise and, in consequence, this will imply a decrease in the price elasticity of demand of public transportation and thus cause a shift in market power. This change in market power should most undoubtedly profit users instead of transport operators. Consequently, this reinforces the idea that the system should be pushed towards a change in how access to opportunities is conceived by operators and planners.

Currently, certain dense networks are already starting to carry out this type of analysis, for example the Access To Opportunities and Services (ATOS) index (COOPER, S. WRIGHT, P. & BALL, R, 2009) proposed by Transport for London in order to improve planning measures. Indeed, just as the UK's planners have begun changing their metrics in order to take into account these new behaviour patterns in transport activities, ITS specialist and planners should seize the opportunity to develop and offer new services. This implies the need to explore new markets that will be based on an economic model taking into account adaptation strategies that conceive the utility of transport users as not only as being a function of opportunities but also of the goods and services that are accessible to them in a reasonable lapse of time [START_REF] Geurs | Accessibility evaluation of land-use and transport strategies: review and research directions[END_REF]. This supports the idea of accompanying changes in the time/cost structure of transport budgets that aim at modifying behaviour with increasing opportunities linked to transport activities and land use patterns :

t time acces time cos )* ( ies opportunit       .

IV. CONCLUDING REMARKS

As public policy evolves, changes in the transport system will suppose behavioural modifications. This situation will face planners and industry deciders with a new challenge: to plan according to a utility function that will not only depend on the opportunities a transport user has but also on the possibility of actually being able to access and consume these opportunities. Indeed, as the transport cost/location structure changes, new trade-offs and optimizations will redefine time use. These changes will have important impacts on the overall macroeconomic structure of a country and will have important effects on the microeconomic choices of transport users. In this sense, sustainable policy planning should be accompanied by adaptation strategies.

Just as transport planning, until now, has built the current transport system around the idea of increasing access to speed (urban motorways, airplanes, high speed rail) in order to gain access to new opportunities [START_REF] Echenique | Mobility and Income[END_REF]& SCHEAFFER, 2009), environmental constraints imply the need to plan for an increase in accessibility without raising speed.

In 1973, BAUMOL, based on LINDER's work, concluded that as productivity grows, free time is pushed towards an increase in the intensity of activities accomplished in a person's free time and goes on to explain that as this happen, time consuming activities will tend to disappear (going to the theatre, or even luxury cruises). In the same year, PHELPS describes consumers as: "a one-man, Mincer-Becker-Lancaster factory" and views commodities as "unfinished materials that need to be combined with the factory's at-home labour time (and sometimes consumer durables) to produce final utility" Although these ideas are nothing new, they seem to be perfectly adapted to what DUJARIER describes as the "next limit of competitiveness". Through this, she explains how firms have faced rising production costs by getting buyers to become co-producers who render their free time productive by creating value whilst participating in the production process of the goods they buy and consume (be it by sheer interest in the activity, because it helps reduce prices or even make-up for forgone earnings in free time).

Just as with any other product, the next limit of transport competitiveness lies in the ability to face rising environmental constraints by increasing the "at-transport time" and the "accesstime" value-intensiveness through time-value management strategies as well as land-use optimization. Although these changes will be difficult for industrialized countries that have built their networks around speed, this seems to be an ideal opportunity for newly industrialized economies to use their already high level of public transport in order to develop new adaptive strategies.

  Indeed, different studies (BANISTER & HICKMAN; KATO & ITO et al; LOPEZ-RUIZ & CROZET; SCHADE & HELFRICH et al; SCHIPPER & NG et al; SPERLING & LUTSEY)