$$C^\infty $$ Smoothing for Weak Solutions of the Inhomogeneous Landau Equation

Henderson, Christopher; Snelson, Stanley

doi:10.1007/s00205-019-01465-7

Cited by 57 publications

(71 citation statements)

References 18 publications

(38 reference statements)

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“…Conditional regularity theory. A thread of recent works concern regularity of solutions to the Landau equation assuming a priori pointwise control of the mass density, energy density and entropy density [32,18,45,46,56]. Our present paper in particular relies on the work [46], which proves the local existence, uniqueness and instantaneous smoothing of solutions using the theory developed in the papers mentioned above.…”

Section: Regularity Theory For Landau Equationmentioning

confidence: 88%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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Consider the spatially inhomogeneous Landau equation with moderately soft potentials (i.e. with γ ∈ (−2, 0)) on the whole space R 3 . We prove that if the initial data fin are close to the vacuum solution fvac ≡ 0 in an appropriate norm, then the solution f remains regular globally in time. This is the first stability of vacuum result for a binary collisional model featuring a long-range interaction.Moreover, we prove that the solutions in the near-vacuum regime approach solutions to the linear transport equation as t → +∞. Furthermore, in general, solutions do not approach a traveling global Maxwellian as t → +∞.Our proof relies on robust decay estimates captured using weighted energy estimates and the maximum principle for weighted quantities. Importantly, we also make use of a null structure in the nonlinearity of the Landau equation which suppresses the most slowly-decaying interactions. * jluk@stanford.edu 1 Each of these terms depends also on (t, x). For brevity, we have suppressed these dependence in (1.2).

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Section: Regularity Theory For Landau Equationmentioning

confidence: 88%

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Luk

2019

Ann. PDE

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“…The regularisation estimate in Hölder spaces was obtained in [30] (third step). The fourth step was completed in [34] in the form of Schauder estimates for kinetic parabolic equations. The regularity of solutions of the Landau equation is iteratively improved using Schauder estimates up to C ∞ regularity.…”

Section: The Boltzmann Equation With Long-range Interactionsmentioning

confidence: 99%

“…The regularity of solutions of the Landau equation is iteratively improved using Schauder estimates up to C ∞ regularity. In the physical case of the Landau-Coulomb equation (playing the role of the limit case α = 2, γ = −3, s = 1 in dimension 3), the conjecture is still open: the L ∞ bound is missing (see however partial results in this direction in [50]), and the Schauder estimates [34] do not cover this case even though this last point is probably only a milder technical issue.…”

Section: The Boltzmann Equation With Long-range Interactionsmentioning

confidence: 99%

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Decay estimates for large velocities in the Boltzmann equation without cutoff

Imbert¹,

Mouhot²,

Silvestre³

2019

Journal De L’École Polytechnique — Mathématiques

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We consider solutions f = f (t, x, v) to the full (spatially inhomogeneous) Boltzmann equation with periodic spatial conditions x ∈ T d , for hard and moderately soft potentials without the angular cutoff assumption, and under the a priori assumption that the main hydrodynamic fields, namely the local mass v f (t, x, v) and local energy v f (t, x, v)|v| 2 and local entropy v f (t, x, v) ln f (t, x, v), are controlled along time. We establish quantitative estimates of propagation in time of "pointwise polynomial moments", i.e. sup x,v f (t, x, v)(1 + |v|) q , q ≥ 0. In the case of hard potentials, we also prove appearance of these moments for all q ≥ 0. In the case of moderately soft potentials we prove the appearance of low-order pointwise moments.

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“…Moreover, all f k will satisfy (16) since f 0 does by (43). Therefore we can apply Proposition 4.1 and conclude that each f k for k from 0 to k 0 must satisfy (18). That means that the set…”

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confidence: 88%

Hölder Continuity for a Family of Nonlocal Hypoelliptic Kinetic Equations

Stokols¹

2019

SIAM J. Math. Anal.

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In this work, Holder continuity is obtained for solutions to the nonlocal kinetic Fokker-Planck Equation, and to a family of related equations with general integro-differential operators. These equations can be seen as a generalization of the Fokker-Planck Equation, or as a linearization of non-cutoff Boltzmann. Difficulties arise because our equations are hypoelliptic, so we utilize the theory of averaging lemmas. Regularity is obtained using De Giorgi's method, so it does not depend on the regularity of initial conditions or coefficients. This work assumes stronger constraints on the nonlocal operator than in the work of Imbert and Silvestre [22], but allows unbounded source terms.2 STOKOLS space with their velocities changing in a stochastic manner. If the velocity of a given particle varied according to the Weiner process, then f would obey a (local) kinetic Fokker-Planck Equation.However, when the velocity of each particle varies according to a Levy process (without drift), the density function obeys (1). A Levy process, unlike the Weiner process, allows individual particles to change velocity suddenly and discontinuously, which better approximates the effect of elastic collisions.Another important model from the statistical mechanics of particles is the Boltzmann Equation). In the non-cutoff case, the Boltzmann Equation sometimes enjoys a regularization effect similar the fractional Fokker-Planck equation (Alexandre, Morimoto, Ukai, Xu, and Yang [2]). Our equation (1) is closely related to the linear approximation of the bilinear collision operator Q(⋅, ⋅). If the mass, energy, and entropy of a solution are assumed to be uniformly bounded, then regularization due to hypoellipticity is observed for the Boltzmann Equation (Imbert and Silvestre [22]), and also for the closely related Landau Equation (Henderson and Snelson [18], Cameron, Silvestre, and Snelson [8]). Note that [22] rewrites the Boltzmann equation in the form (1), but with kernel satisfying weaker constraints than (2). Their regularity results are discussed below. The most important assumption these papers require is that the mass is bounded away from the vacuum, which is connected to the coercivity of the collision operator. In [19], Henderson, Snelson, and Tarfulea show that this assumption really does hold for the Landau Equation. See Mouhot [30] for a thorough review of the current state of research on this front.Equation (1) is a typical hypoelliptic equation. Although regularization of the integral operator happens only in v, we will gain regularity in t, x thanks to the mixing property of the transport operator. This is reminiscent of the hypoelliptic theory based on C ∞ of Hörmander [20] and Kolmogorov. Averaging lemmas such as [14] (Golse, Lions, Perthame, Sentis) can be seen as an H s theory of hypoellipticity.This H s theory has already been applied specifically to the nonlocal kinetic Fokker-Planck Equation. Lerner, Morimoto, showed that solutions to certain fractional kinetic equations are in a Sobolev space H σ in all three variable...

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$$C^\infty $$ Smoothing for Weak Solutions of the Inhomogeneous Landau Equation

Cited by 57 publications

References 18 publications

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Stability of Vacuum for the Landau Equation with Moderately Soft Potentials

Decay estimates for large velocities in the Boltzmann equation without cutoff

Hölder Continuity for a Family of Nonlocal Hypoelliptic Kinetic Equations

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