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.
For the homogeneous Landau equation, which arises when @math is assumed to be independent of @math in , the theory is more developed. The @math smoothing is established for hard potentials in @cite_3 and for Maxwell molecules in @cite_5 , under the assumption that the initial data has finite mass and energy. Propagation of @math estimates in the case of moderately soft potentials was shown in @cite_0 and @cite_4 . Global upper bounds in a weighted @math space were established in @cite_20 , even for @math , as a consequence of entropy dissipation. Global @math bounds that do not depend on @math and that do not degenerate as @math were derived in @cite_21 for moderately soft potentials, and this result also implies @math smoothing by standard parabolic regularity theory. Note that in the space homogeneous case our assumptions , and hold for all @math provided that the initial data has finite mass, energy and entropy. Both Theorems and are new results even in the space homogeneous case. The previous results for soft potentials do not address the decay of the solution for large velocities.
@cite_4 @cite_21 @cite_3 @cite_0 @cite_5 @cite_20
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This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in @math space hold true for moderately soft potential cases @math without any smallness assumption on the initial data. For very soft potential cases @math , which cover in particular the Coulomb case @math , we get local in time estimates of weak solutions in @math . In the proofs of these estimates, global ones for the special case @math and local ones for very soft potential cases @math , the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality. Abstract We consider a parabolic equation in nondivergence form, defined in the full space [ 0 , ∞ ) × R d , with a power nonlinearity as the right-hand side. We obtain an upper bound for the solution in terms of a weighted control in L p . This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in L ∞ ( R d ) for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data. We study the Cauchy problem for the homogeneous Landau equation of kinetic theory, in the case of hard potentials. We prove that for a large class of initial data, there exists a unique weak solution to this problem, which becomes immediately smooth and rapidly decaying at infinity. 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. We establish a simplified form for the Landau equation with Maxwellian-type molecules. We study in details the Cauchy problem associated to this equation, and some qualitative features of the solution. Explicit solutions are given. Abstract We present in this paper an estimate which bounds from below the entropy dissipation D ( f ) of the Landau operator with Coulomb interaction by a weighted H 1 norm of the square root of f . As a consequence, we get a weighted L t 1 ( L v 3 ) estimate for the solutions of the spatially homogeneous Landau equation with Coulomb interaction, and the propagation of L 1 moments of any order for this equation. We also present an application of our estimate to the Landau equation with (moderately) soft potentials, providing thus a new proof of some recent results of [30] .