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.
In order to keep track of the constants for parabolic regularization estimates (as in @cite_22 ) for large velocities, we describe a change of variables in Lemma . This change of variables may be useful in other contexts. It is related to one mentioned in the appendix of @cite_17 for the Boltzmann equation.
@cite_22 @cite_17
230142061 1989465898 2156027049 2045533739
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Holder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in ( L^ ) which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded. 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. The time-dependent Boltzmann equation, which describes the propagation of radiation from a point source in a random medium, is solved exactly in Fourier space. An explicit expression in real space is given in two and four dimensions. In three dimensions an accurate interpolation formula is found. The average intensity at a large distance @math from the source has two peaks, a ballistic peak at time @math and a diffusion peak at @math (with @math the velocity and @math the diffusion coefficient). We find that forward scattering adds a tail to the ballistic peak in two and three dimensions, @math and @math , respectively. Expressions in the literature do not contain this tail. In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular:---We show that, for any p ∈ (0, 2], one can maintain (using only O(log n e2) words of storage) a sketch C(q) of a point q ∈ lnp under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate Vq − sVp up to a factor of (1 p e) with large probability. This solves the main open problem of [1999].---We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ ℜn and y ∈ ℜm, estimates VAx − yVp in O(n p m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p e 2.---We obtain another sketch function C′ which probabilistically embeds ln1 into a normed space lm1. The embedding guarantees that, if we set m e log(1 Δ)O(1 e), then for any pair of points q, s ∈ ln1, the distance between q and s does not increase by more than (1 p e) with constant probability, and it does not decrease by more than (1 − e) with probability 1 − Δ. This is the only known dimensionality reduction theorem for the l1 norm. In fact, stronger theorems of this type (i.e., that guarantee very low probability of expansion as well as of contraction) cannot exist [Brinkman and Charikar 2003].---We give an explicit embedding of ln2 into lnO(log n)1 with distortion (1 p 1 nΘ(1)).