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Register a new userPropagation of Gevrey regularity for solutions of the Boltzmann equation for Maxwellian molecules
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We prove that Gevrey regularity is propagated by the Boltzmann equation with Maxwellian molecules, with or without angular cut-off. The proof relies on the Wild expansion of the solution to the equation and on the characterization of Gevrey regularity by the Fourier transform.
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We show that convex viscosity solutions of the Lagrangian mean curvature equation are regular if the Lagrangian phase has Holder continuous second derivatives.
In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain $mathbb{R}^d$, $dgeq 2$, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $left< v right>^beta f_0 in L^2_v H^alpha_x$ for $min (alpha,beta) > frac{d-1}{2}$. We prove that if $alpha,beta$ are large enough, then it is possible to propagate moments in $x$ and derivatives in $v$ (for instance, $left< x right>^k left< abla_v right>^ell f in L^infty_T L^2_{x,v}$ if $f_0$ is nice enough). The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of $f$. We also prove a persistence of regularity result for the scale of Sobolev spaces $H^{alpha,beta}$; and, continuity of the solution map in $H^{alpha,beta}$. Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the $H$-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in $H^{alpha,beta}$ for any $alpha,beta > frac{d-1}{2}$, without any additional regularity or decay assumptions.
The well-known Rutherford differential cross section, denoted by $ dOmega/dsigma$, corresponds to a two body interaction with Coulomb potential. It leads to the logarithmically divergence of the momentum transfer (or the transport cross section) which is described by $$int_{{mathbb S}^2} (1-costheta) frac{dOmega}{dsigma} dsigmasim int_0^{pi} theta^{-1}dtheta. $$ Here $theta$ is the deviation angle in the scattering event. Due to screening effect, physically one can assume that $theta_{min}$ is the order of magnitude of the smallest angles for which the scattering can still be regarded as Coulomb scattering. Under ad hoc cutoff $theta geq theta_{min}$ on the deviation angle, L. D. Landau derived a new equation in cite{landau1936transport} for the weakly interacting gas which is now referred to as the Fokker-Planck-Landau or Landau equation. In the present work, we establish a unified framework to justify Landaus formal derivation in cite{landau1936transport} and the so-called Landau approximation problem proposed in cite{alexandre2004landau} in the close-to-equilibrium regime. Precisely, (i). we prove global well-posedness of the Boltzmann equation with cutoff Rutherford cross section which is perhaps the most singular kernel both in relative velocity and deviation angle. (ii). we prove a global-in-time error estimate between solutions to Boltzmann and Landau equations with logarithm accuracy, which is consistent with the famous Coulomb logarithm. Key ingredients into the proofs of these results include a complete coercivity estimate of the linearized Boltzmann collision operator, a uniform spectral gap estimate and a novel linear-quasilinear method.
We study the positivity and regularity of solutions to the fractional porous medium equations $u_t+(-Delta)^su^m=0$ in $(0,infty)timesOmega$, for $m>1$ and $sin (0,1)$ and with Dirichlet boundary data $u=0$ in $(0,infty)times({mathbb R}^NsetminusOmega)$, and nonnegative initial condition $u(0,cdot)=u_0geq0$. Our first result is a quantitative lower bound for solutions which holds for all positive times $t>0$. As a consequence, we find a global Harnack principle stating that for any $t>0$ solutions are comparable to $d^{s/m}$, where $d$ is the distance to $partialOmega$. This is in sharp contrast with the local case $s=1$, in which the equation has finite speed of propagation. After this, we study the regularity of solutions. We prove that solutions are classical in the interior ($C^infty$ in $x$ and $C^{1,alpha}$ in $t$) and establish a sharp $C^{s/m}_x$ regularity estimate up to the boundary. Our methods are quite general, and can be applied to a wider class of nonlocal parabolic equations of the form $u_t-mathcal L F(u)=0$ in $Omega$, both in bounded or unbounded domains.
This paper discusses some regularity of almost periodic solutions of the Poissons equation $-Delta u = f$ in $mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poissons equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poissons equation in the distribution sense, then $u$ is an almost periodic function. In this work, we relax the assumption of the usual boundedness into boundedness in the sense of distribution which we refer to as a bounded generalized function. The set of bounded generalized functions are wider than the set of usual bounded functions. Then, upon assuming that $u$ is a bounded generalized function and solves the Poissons equation in the distribution sense, we prove that this solution is bounded in the usual sense, continuous and almost periodic. Moreover, we show that the first partial derivatives of the solution $partial u/ partial x_i$, $i=1, ldots, n$, are also continuous, bounded, and almost periodic functions. The technique is based on extending a representation formula using Greens function for Poissons equation for solutions in the distribution sense. Some useful properties of distributions are also shown that can be used to study other elliptic problems.
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