We consider the spatially inhomogeneous Landau equation with initial data that is bounded by a Gaussian in the velocity variable. In the case of moderately soft potentials, we show that weak solutions immediately become smooth and remain smooth as long as the mass, energy, and entropy densities remain under control. For very soft potentials, we obtain the same conclusion with the additional assumption that a sufficiently high moment of the solution in the velocity variable remains bounded. Our proof relies on the iteration of local Schauder-type estimates.
Regarding the existence theory for , global-in-time classical solutions have only been constructed in the close-to-equilibrium setting: see the work of Guo @cite_7 in the @math -periodic case, and Mouhot-Neumann @cite_0 in the whole space. For general initial data, Villani @cite_15 constructed so-called renormalized solutions with defect measure for the Landau equation. More recently, He-Yang @cite_20 established the short-time existence of spatially periodic classical solutions to in the Coulomb case ( @math ) with initial data in a weighted @math space, by taking the grazing collisions limit in their estimates on the Boltzmann collision operator. They assume that the mass density of the initial data is uniformly bounded away from zero. Since this lower bound along with the bounds , , , and can be shown to propagate for a short time, our Theorem combined with @cite_20 provides a @math solution to the Cauchy problem for suitable initial data. However, on physical grounds, the equation should be expected to be well-posed even with vacuum regions in the initial data. We explore this issue, as well as short-time existence for a broader range of @math , in a forthcoming paper.
@cite_0 @cite_15 @cite_20 @cite_7
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We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc. In the present work, we consider the well-posedness and asymptotics of grazing collisions limit of the spatially inhomogeneous Boltzmann equation with Coulomb interaction. Under the screening hypothesis on the cross-section @math of the Boltzmann collision operator, that is, @math , where @math , we prove that there exists a common lifespan @math such that for any @math , the Boltzmann equation admits a unique nonnegative solution @math in the function space @math when the initial data belongs to the weighted Sobolev space @math with @math and @math . Moreover, it is proved that the solution @math is uniformly bounded with respect to the parameter @math in the above function spaces. As a consequence, by using the ... Abstract This paper studies the approximation of the Boltzmann equation by the Landau equation in a regime when grazing collisions prevail. While all previous results in the subject were limited to the spatially homogeneous case, here we manage to cover the general, space-dependent situation, assuming only basic physical estimates of finite mass, energy, entropy and entropy production. The proofs are based on the recent results and methods introduced previously in [R. Alexandre, C. Villani, Comm. Pure Appl. Math. 55 (1) (2002) 30–70] by both authors, and the entropy production smoothing effects established in [R. , Arch. Rational Mech. Anal. 152 (4) (2000) 327–355]. We are able to treat realistic singularities of Coulomb type, and approximations of the Debye cut. However, our method only works for finite-time intervals, while the Landau equation is supposed to describe long-time corrections to the Vlasov–Poisson equation. If the mean-field interaction is neglected, then our results apply to physically relevant situations after a time rescaling. In this paper we study the large-time behavior of classical solutions to the two-species Vlasov-Maxwell-Boltzmann system in the whole space amssym @math . The existence of global-in-time nearby Maxwellian solutions is known from Strain in 2006. However, the asymptotic behavior of these solutions has been a challenging open problem. Building on our previous work on time decay for the simpler Vlasov-Poisson-Boltzmann system, we prove that these solutions converge to the global Maxwellian with the optimal decay rate of O(t−3 2 + 3 (2r)) in the L (L)-norm for any 2 ≤ r ≤ ∞ if initial perturbation is smooth enough and decays in space velocity fast enough at infinity. Moreover, some explicit rates for the electromagnetic field tending to 0 are also provided. © 2011 Wiley Periodicals, Inc.