We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under control. Our pointwise estimates decay polynomially in the velocity variable. We also show that if the initial data satisfies a Gaussian upper bound, this bound is propagated for all positive times.
Classical solutions for have so far only been constructed in a close-to-equilibrium setting: see the work of Guo @cite_7 and Mouhot-Neumann @cite_1 . A suitable notion of weak solution, for general initial data, was constructed by Alexandre-Villani @cite_15 @cite_14 .
@cite_15 @cite_14 @cite_1 @cite_7
2135329241 1509009208 1996180590 1983529651
Abstract This paper deals with some global in time a priori estimates of the spatially homogeneous Landau equation for soft potentials γ ∈ [ − 2 , 0 ) . For the first result, we obtain the estimate of weak solutions in L t α L v 3 − e for α = 2 ( 3 − e ) 3 ( 2 − e ) and 0 e 1 , which is an improvement over estimates by Fournier and Guerin [10] . For the second result, we have the estimate of weak solutions in L t ∞ L v p , p > 1 , which extends part of results by Fournier and Guerin [10] and Alexandre, Liao and Lin [1] . As an application, we deduce some global well-posedness results for γ ∈ [ − 2 , 0 ) . Our estimates include the case γ = − 2 , which is the key point in this paper. This article proposes a solution of the Lambertian Shape From Shading (SFS) problem by designing a new mathematical framework based on the notion of viscosity solutions. The power of our approach is twofolds: 1) it defines a notion of weak solutions (in the viscosity sense) which does not necessarily require boundary data. Note that, in the previous SFS work of [23,15], [8], [22,20], the characterization of a viscosity solution and its computation require the knowledge of its values on the boundary of the image. This was quite unrealistic because in practice such values are not known. 2) it unifies the work of [23,15], [8], [22,20], based on the notion of viscosity solutions and the work of Dupuis and Oliensis [6] dealing with classical (C 1) solutions. Also, we generalize their work to the “perspective SFS” problem recently introduced by Prados and Faugeras 20. We consider variants of the classical stable marriage problem in which preference lists may contain ties, and may be of bounded length. Such restrictions arise naturally in practical applications, such as centralised matching schemes that assign graduating medical students to their first hospital posts. In such a setting, weak stability is the most common solution concept, and it is known that weakly stable matchings can have different sizes. This motivates the problem of finding a maximum cardinality weakly stable matching, which is known to be NP-hard in general. We show that this problem is solvable in polynomial time if each man's list is of length at most 2 (even for women's lists that are of unbounded length). However if each man's list is of length at most 3, we show that the problem becomes NP-hard (even if each women's list is of length at most 3) and not approximable within some @d>1 (even if each woman's list is of length at most 4). We adapt the iterative scheme by Moser, to prove that the weak solutions to an ultraparabolic equation, with measurable coefficients, are locally bounded functions. Due to the strong degeneracy of the equation, our method differs from the classical one in that it is based on some ad hoc Sobolev type inequalities for solutions.