Boundaries occur naturally in kinetic equations and boundary effects are crucial for dynamics of dilute gases governed by the Boltzmann equation. We develop a mathematical theory to study the time decay and continuity of Boltzmann solutions for four basic types of boundary conditions: inflow, bounce-back reflection, specular reflection, and diffuse reflection. We establish exponential decay in $L^{\infty}$ norm for hard potentials for general classes of smooth domains near an absolute Maxwellian. Moreover, in convex domains, we also establish continuity for these Boltzmann solutions away from the grazing set of the velocity at the boundary. Our contribution is based on a new $L^{2}$ decay theory and its interplay with delicate $% L^{\infty}$ decay analysis for the linearized Boltzmann equation, in the presence of many repeated interactions with the boundary.
We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its Hölder continuity at regular free boundary points.
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and its applications to strongly coupled systems does not require higher $L^p$ integrability of derivatives.
In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with three iterations, we establish regularity in fractional Sobolev space in space variable up to order $1^-$.
We investigate the Boltzmann equation with spatial smearing, diffusive boundary conditions, and Lions' collision kernel. Both, the physical as well as the velocity space, are assumed to be bounded. Existence and uniqueness of a stationary solution, which is a probability density, has been demonstrated in [3] under a certain smallness assumption on the collision term. We prove that whenever there is a stationary solution then it is a.e. positively bounded from below and above.
This paper is concerned with the Boltzmann equation with specular reflection boundary condition. We construct a unique global solution and obtain its large time asymptotic behavior in the case that the initial data is close enough to a radially symmetric homogeneous datum. The result extends the case of Cauchy problem considered by Arkeryd-Esposito-Pulvirenti [Comm. Math. Phys. 111(3): 393-407 (1987)] to the specular reflection boundary value problem in bounded domain.
In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in $L^2_{x,v}\cap L^\infty_{x,v}$ in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.
We prove two results of strong continuity with respect to the initial datum for bounded solutions to the Euler equations in vorticity form. The first result provides sequential continuity and holds for a general bounded solution. The second result provides uniform continuity and is restricted to Hölder continuous solutions.
This article is about the convex solution $u$ of the Monge--Ampère equation on an at least 2-dimensional open bounded convex domain with Dirichlet boundary data and nonnegative bounded right-hand side. For convex functions with zero boundary data, an Alexandrov maximum principle $|u(x)| \leq C \operatorname{dist}(x,\partialΩ)^α$ is equivalent to (uniform) Hölder continuity with the same constant and exponent. Convex $α$-Hölder continuous functions are $W^{1,p}$ for $p < 1/(1{-}α)$. We prove Hölder continuity with the exponent $α=2/n$ for $n \geq 3$ and any $α\in (0,1)$ for $n=2$, provided that…
Locally bounded, local weak solutions to a special class of quasilinear, anisotropic, $p$-Laplacian type elliptic equations, are shown to be locally Hölder continuous. Homogeneous local upper bounds are established for local weak solutions to a general class of quasilinear anisotropic equations.
Let $λ_{q}:=\inf{\Vert\nabla u\Vert_{L^{p}(Ω)}^{p}/\Vertu\Vert_{L^{q}(Ω)}^{p}:u\in W_{0}^{1,p}(Ω)\setminus{0}} $, where $Ω$ is a bounded and smooth domain of $\mathbb{R}^{N},$ $1<p<N$ and $1\leq q\leq p^{\star}% :=\frac{Np}{N-p}.$ We prove that the function $q\mapstoλ_{q}$ is absolutely continuous in the closed interval $[1,p^{\star}].$
We establish a Riesz potential criterion for Lebesgue continuity points of functions of bounded $\mathbb{A}$-variation, where $\mathbb{A}$ is a $\mathbb{C}$-elliptic differential operator of arbitrary order. This result might even be of interest for classical functions of bounded variation.
We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional $p$-Laplacian operator on the Heisenberg-Weyl group $\mathbb{H}^n$. Amongst other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates.
Consider 3D Boltzmann equation in convex domains with diffusive-reflection boundary condition. We study the hydrodynamic limits as the Knudsen number and Strouhal number $ε\rightarrow 0^+$. Using the Hilbert expansion, we rigorously justify that the solution of stationary/evolutionary problem converges to that of the steady/unsteady Navier-Stokes-Fourier system. This is the first paper to justify the hydrodynamic limits of nonlinear Boltzmann equations with hard-sphere collision kernel in bounded domain in the $L^{\infty}$ sense. The proof relies on a novel analysis on the boundary layer effec…
We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild…
In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao [Nonlinear Anal. 36 (1999), no. 3, 295--318.] and Winkert-Zacher [Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865--878.] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique…
When dilute charged particles are confined in a bounded domain, boundary effects are crucial in the global dynamics. We construct a unique global-in-time solution to the Vlasov-Poisson-Boltzmann system in convex domains with the diffuse boundary condition. The construction is based on an $L^{2}$-$L^{\infty}$ framework with a novel \textit{nonlinear-normed energy estimate} of a distribution function in weighted $W^{1,p}$-spaces and a $C^{2,δ}$-estimate of the self-consistent electric potential. Moreover we prove an exponential convergence of the distribution function toward the global Maxwellia…
We investigate the regularity issue for the diffuse reflection boundary problem to the stationary linearized Boltzmann equation for hard sphere potential, cutoff hard potential, or cutoff Maxwellian molecular gases in a strictly convex bounded domain. We obtain pointwise estimates for first derivatives of the solution provided the boundary temperature is bounded differentiable and the solution is bounded. This result can be understood as a stationary version of the velocity averaging lemma and mixture lemma.
We study steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions and the source term decaying exponentially, we establish the existence of boundary layer solution for both linear and nonlinear Boltzmann equation in half-space with diffusive reflection boundary condition in $L^\infty_{x,v}$ when the far-field Mach number of the Maxwellian is zero. The continuity and the spacial decay of the solution are obtained. The uniqueness is established under some constraint conditions.
Let $Ω\subset\mathbb{R}^{2}$ be a bounded, Lipschitz domain. We consider bounded, weak solutions ($u\in W^{1, 2}\cap L^{\infty}(Ω;\mathbb{R}^N)$) of the vector-valued, Euler-Lagrange system: \text{div } \big( A(x, u)Du\big)=g(x, u, Du)\quad\text{in }Ω. Under natural growth conditions on the principal part and the inhomogeneity, but without any further restriction on the growth of the inhomogeneity (for example, via a smallness condition), we use a blow-up argument to prove that every bounded, weak solution of the system is Hölder continuous. Since the dimension of $Ω$ is $2$ and $u\in W^{1, 2}…
Boundary effects are crucial for dynamics of dilute charged gases governed by the Vlasov-Poisson-Boltzmann (VPB) system. In this paper, we study the existence and regularity of solutions to the VPB system with soft potential in a bounded convex domain with in-flow boundary condition. We establish the existence of strong solutions in the time interval $[0,T]$ for an arbitrary given $T>0$ when the initial distribution function is near an absolute Maxwellian. Our contribution is based on a new weighted energy estimate in some $W^{1,p}$ space and $L_x^3 L_v^{1+}$ space for soft potential. By using…
We prove the existence of global renormalized solutions to the Boltzmann equation in bounded domain with incoming boundary condition, with non-cutoff collision kernels. Thus we extend the results of \cite{villani2002noncutoff} for whole spaces or periodic domain to bounded domains endorsed with incoming boundary condition.
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 nonloca…
Here we generalize quasilinear parabolic $p-$Laplacian type equations to obtain the prototype equation as \[ u_t - \text{div} (g(|Du|)/ |Du| \cdot Du) = 0, \] where a nonnegative, increasing, and continuous function $g$ trapped in between two power functions $|Du|^{g_0 -1}$ and $|Du|^{g_1 -1}$ with $1<g_0 \leq g_1 < \infty$. Through this generalization in the setting from Orlicz spaces, we provide a uniform proof with a single geometric setting that a bounded weak solution is locally Hölder continuous considering $1 < g_0 \leq g_1 \leq 2$ and $2 \leq g_0 \leq g_1 < \infty$ separately. By using…
It is well known that the full compressible Navier-Stokes equations can be deduced via the Chapman-Enskog expansion from the Boltzmann equation as the first-order correction to the Euler equations with viscosity and heat-conductivity coefficients of order of the Knudsen number $ε>0$. In the paper, we carry out the rigorous mathematical analysis of the compressible Navier-Stokes approximation for the Boltzmann equation regarding the initial-boundary value problems in general bounded domains. The main goal is to measure the uniform-in-time deviation of the Boltzmann solution with diffusive refle…
The existence and stability of collisional kinetic equation, especially non-cutoff Boltzmann equation, in bounded domain with physical boundary condition is longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in bounded domain. Our method is based on that fact that normal derivatives in cubes is also derivatives along axis, which allows us to obtain high-order …
For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in the whole space $\mathbb R^3_x$, we establish pointwise lower bounds that appear instantaneously even if the initial data contains vacuum regions. Our lower bounds depend only on the initial data and upper bounds for the mass and energy densities of the solution. As an application, we improve the weakest known continuation criterion for large-data solutions, by removing the assumptions of mass bounded below and entropy bounded above.
The paper is devoted to constructing the global solutions around global Maxwellians to the initial-boundary value problem on the Boltzmann equation in general bounded domains with isothermal diffuse reflection boundaries. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in $L^2$. This yields the disappearance of any initial vacuum and the exponential convergence of l…
We study generalized fractional $p$-Laplacian equations to prove local boundedness and Hölder continuity of weak solutions to such nonlocal problems by finding a suitable fractional Sobolev-Poincaré inquality.
We characterize probability measure with finite moment of any order in terms of the symmetric difference operators of their Fourier transforms. By using our new characterization, we prove the continuity $f(t,v)\in C((0, \infty),L^1_{2k-2 +α})$, where $f(t, v)$ stands for the density of unique measure-valued solution $(F_t)_{t\ge0}$ of the Cauchy problem for the homogeneous non-cutoff Boltzmann equation, with Maxwellian molecules, corresponding to a probability measure initial datum $F_0$ satisfying \[ \int |v|^{2k-2+α} dF_0(v) < \infty, 0\leq α< 2,k= 2, 3, 4,\cdots \] provided that $F_0$ is no…
We establish pointwise polynomial decay estimates in velocity space for the spatially inhomogeneous Boltzmann equation without cutoff, in the case of hard potentials ($γ+2s > 2$), under the assumption that the mass, energy, and entropy densities are bounded above, and the mass density is bounded below. These estimates are self-generating, i.e. they do not require corresponding decay assumptions on the initial data. Our results extend the recent work of Imbert-Mouhot-Silvestre (arXiv:1804.06135), which addressed the case of moderately soft potentials ($γ+ 2s \in [0,2]$).
In this paper, we obtain the weak Harnack inequality and Hölder estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cut-off can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local Hölder estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cut-off.
We prove that $L^2$ weak solutions to hypoelliptic equations with bounded measurable coefficients are Hölder continuous. The proof relies on classical techniques developed by De Giorgi and Moser together with the averaging lemma and regularity transfers developed in kinetic theory. The latter tool is used repeatedly: first in the proof of the local gain of integrability of sub-solutions; second in proving that the gradient with respect to the velocity variable is $L^{2+ε}_{\mathrm{loc}}$; third, in the proof of an "hypoelliptic isoperimetric De Giorgi lemma." To get such a lemma, we develop a …
Locally bounded, local weak solutions to a doubly nonlinear parabolic equation, which models the multi-phase transition of a material, is shown to be locally continuous. Moreover, an explicit modulus of continuity is given. The effect of the $p$-Laplacian type diffusion is also considered.
We discuss some recent development on the Vlasov-Poisson-Boltzmann system in bounded domains with diffuse reflection boundary condition. In addition we present a new regularity result when the particles are surrounded by conductor boundary.
Conditional appearance of decay for the non-cutoff Boltzmann equation in a bounded domain
Département de Mathématiques et Applications, ENS（高等师范学院数学与应用系）；Department of Pure Mathematics and Mathematical Statistics, University of Cambridge（剑桥大学纯数学与数理统计系）
In this brief note we discuss local Hölder continuity for solutions to anisotropic elliptic equations of the type $ \sum_{i=1}^s \partial_{ii} u+ \sum_{i=s+1}^N \partial_i \bigg(A_i(x,u,\nabla u) \bigg) =0,$ for $x \in Ω\subset \subset \mathbb{R}^N$ and $1\leq s \leq N-1$, where each operator $A_i$ behaves directionally as the singular $p$-Laplacian, $1< p < 2$ and the supercritical condition $p+(N-s)(p-2)>0$ holds true. We show that the Harnack inequality can be proved without the continuity of solutions and that in turn this implies Hölder continuity of solutions.
We show that bounded solutions of the quasilinear elliptic equation $Δ_{p(x)} u=g+div(\textbf{F})$ are locally Hölder continuous provided that the functions $g$ and $\textbf{F}$ are in suitable Lebesgue spaces.
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
This note gives a first sufficient condition that insures a non-negative, locally bounded, local solution to a logarithmically singular parabolic equation is continuous at a vanishing point and an estimate of the modulus of continuity is given. Moreover, an estimate of the Hausdorff measure of the set of discontinuity is established.
We study local boundedness and Hölder continuity of a parabolic equation involving the fractional $p$-Laplacian of order $s$, with $0<s<1$, $2\leq p < \infty$, with a general right hand side. We focus on obtaining precise Hölder continuity estimates. The proof is based on a perturbative argument using the already known Hölder continuity estimate for solutions to the equation with zero right hand side.
We consider certain Boltzmann type equations on a bounded physical and a bounded velocity space under the presence of both, reflective as well as diffusive boundary conditions. We introduce conditions on the shape of the physical space and on the relation between the reflective and the diffusive part in the boundary conditions such that the associated Knudsen type semigroup can be extended to time $t\in\mathbb{R}$. Furthermore, we provide conditions under which there exists a unique global solution to a Boltzmann type equation for time $t\ge 0$ or for time $t\in [τ_0,\infty)$ for some $τ_0<0$ …
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi an…
We study the weak continuity of two interrelated non-linear partial differential equations, the Yang-Mills equations and the Gauß-Codazzi-Ricci equations, involving $L^p$-integrable connections. Our key finding is that underlying cancellations in the curvature form, especially the div-curl structure inherent in both equations, are sufficient to pass to the limit in the non-linear terms. We first establish the weak continuity of Yang-Mills equations and prove that any weakly converging sequence of weak Yang-Mills connections in $L^p$ converges to a weak Yang-Mills connection. We then prove that…
In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in \cite{Caflisch 97}, and the boundedness and the Besov-type continuity for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that the Besov-type continuity condition near the boundary is consistent with the interior Besov regularity, which is a ne…
We study the incompressible Euler equations in the $α$-modulation $M^{s,α}_{p,q}$ and Hölder $C^{1+σ}$ spaces on the plane. We show that for these spaces the associated data-to-solution map is not continuous on bounded sets.
The Boltzmann equation with non-isothermal Maxwell boundary conditions
Centre de Recherche en Mathématiques de la Décision (CEREMADE, CNRS UMR 7534), Universités PSL & Paris-Dauphine（决策数学研究中心（CEREMADE，CNRS UMR 7534），PSL大学与巴黎第九大学）
We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo in $L^\infty_{x,v}\left(\left(1+|v|\right)^βe^{|v|^2/4}\right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.
We consider the higher order Schrödinger operator $H=(-Δ)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$, $m>1$. When $n$ is odd, we prove that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for all $1\leq p\leq\infty$ under $n$ and $m$ dependent conditions on the potential analogous to the case when $m=1$. Further, if $V$ is small in certain norms, that depend $n$ and $m$, the wave operators are bounded on the same range for even $n$. We further show that if the smallness assumption is removed in even dimensions the wave operators …
