In this paper we produce new, optimal, regularity results for the solutions to $p$-Poisson equations. We argue through a delicate approximation method, under a smallness regime for the exponent $p$, that imports information from a limiting profile driven by the Laplace operator. Our arguments contain a novelty of technical interest, namely a sequential stability result; it connects the solutions to $p$-Poisson equations with harmonic functions, yielding improved regularity for the former. Our findings relate a smallness regime with improved $\mathcal{C}^{1,1-}$-estimates in the presence of $L^\infty$-source terms.
The SBV regularity of weak entropy solutions to the Burgers-Poisson equation is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.
We prove a local regularity result for distributional solutions of the Poisson's equation with $L^p$ data. We use a very short argument based on Weyl's lemma and Riesz-Fréchet representation theorem.
We study the regularity of the $p$-Poisson equation $$ Δ_p u = h, \quad h\in L^q $$ in the plane. In the case $p>2$ and $2<q<\infty$ we obtain the sharp Hölder exponent for the gradient. In the other cases we come arbitrarily close to the sharp exponent.
In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions.
We establish new quantitative estimates for localized finite differences of solutions to the Poisson problem for the fractional Laplace operator with homogeneous Dirichlet conditions of solid type settled in bounded domains satisfying the Lipschitz cone regularity condition. We then apply these estimates to obtain (i)~regularity results for solutions of fractional Poisson problems in Besov spaces; (ii)~quantitative stability estimates for solutions of fractional Poisson problems with respect to domain perturbations; (iii)~quantitative stability estimates for eigenvalues and eigenfunctions of f…
In this paper, inspired by the seminal work by Caffarelli-Kohn-Nirenberg \cite{CKN} on the incompressible Navier-Stokes equation, we establish the existence of a suitable weak solution to the Navier-Stokes-Planck-Nernst-Poisson equation in dimension three, which is shown to be smooth away from a closed set whose $1$-dimensional parabolic Hausdorff measure is zero.
In this paper we study the regularity of the non-cutoff Vlasov-Poisson-Boltzmann system for plasma particles of two species in the whole space $\mathbb{R}^3$ with hard potential. The existence of global-in-time nearby Maxwellian solutions is known for soft potential from [15]. However the smoothing effect of these solutions has been a challenging open problem. We establish the global existence and regularizing effect to the Cauchy problem for hard potential with large time decay. Hence, the solutions are smooth with respect to $(t,x,v)$ for any positive time $t>0$. This gives the regularity to…
We show propagation of moments in velocity for the 3-dimensional Vlasov-Poisson system with a uniform magnetic field $B = (0, 0, ω)$ by adapting the work of Lions, Perthame. The added magnetic field also produces singularities at times which are the multiples of the cyclotron period $t = \dfrac{2πk}ω, k \in \mathbb{N}$ . This result also allows to show propagation of regularity for the solution. For uniqueness, we extend Loeper's result by showing that the set of solutions with bounded macroscopic density is a uniqueness class.
This paper is devoted to the investigation of the boundary regularity for the Poisson equation $${{cc} -Δu = f & \text{in} Ω u= 0 & \text{on} \partial Ω$$ where $f$ belongs to some $L^p(Ω)$ and $Ω$ is a Reifenberg-flat domain of $\mathbb R^n.$ More precisely, we prove that given an exponent $α\in (0,1)$, there exists an $\varepsilon>0$ such that the solution $u$ to the previous system is locally Hölder continuous provided that $Ω$ is $(\varepsilon,r_0)$-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman's monotonicity formula and Morrey-Campanato theorem.
In this paper we consider the Euler-Poisson system (describing a plasma made of ions with a negligible ion temperature) on the Sobolev spaces $H^s(\R^3)$, $s > 5/2$. Using a geometric approach we show that for any time $T > 0$ the corresponding solution map, $(ρ_0,u_0) \mapsto (ρ(T),u(T))$, is nowhere locally uniformly continuous. On the other hand it turns out that the trajectories of the ions are analytic curves in $\R^3$.
We study a basic plasma physics model--the one-fluid Euler--Poisson system on the square torus, in which a compressible electron fluid flows under its own electrostatic field. In this paper we prove long-term regularity of periodic solutions of this system in 2 spatial dimensions. Our main conclusion is that on a square torus of side length $R$, if the initial data is sufficiently close to a constant solution, then the solution is wellposed for a time at least $R/ε^2(\log R)^{O(1)}$, where $ε$ is the size of the initial data.
We study the regularity of solutions of the Poisson equation with Dirichlet, Neumann and mixed boundary values in polyhedral cones $K\subset \mathbb{R}^3$ in the specific scale $\ B^α_{τ,τ}, \ \frac{1}τ=\fracα{d}+\frac{1}{p}\ $ of Besov spaces. The regularity of the solution in these spaces determines the order of approximation that can be achieved by adaptive and nonlinear numerical schemes. We aim for a thorough discussion of homogeneous and inhomogeneous boundary data in all settings studied and show that the solutions are much smoother in this specific Besov scale compared to the fractiona…
We prove propagation of $\frac{1}{s}$-Gevrey regularity $(s\in(0,1))$ for the Vlasov-Poisson system on $\mathbb{T}^d$ using a Fourier space method in analogy to the results proved for the 2D Euler system in \cite{KV} and \cite{LO}. More precisely, we give a quantitative estimate for the growth in time of the $\frac{1}{s}$-Gevrey norm for the solution of the system in terms of the force field and the gradient in the velocity variable of the distribution of matter. As an application, we show global existence of $\frac{1}{s}$-Gevrey solutions ($s\in (0,1)$) for the Vlasov-Poisson system in $\math…
We consider the normalized $p$-Poisson problem $$-Δ^N_p u=f \qquad \text{in}\quad Ω.$$ The normalized $p$-Laplacian $Δ_p^{N}u:=|D u|^{2-p}Δ_p u$ is in non-divergence form and arises for example from stochastic games. We prove $C^{1,α}_{loc}$ regularity with nearly optimal $α$ for viscosity solutions of this problem. In the case $f\in L^{\infty}\cap C$ and $p>1$ we use methods both from viscosity and weak theory, whereas in the case $f\in L^q\cap C$, $q>\max(n,\frac p2,2)$, and $p>2$ we rely on the tools of nonlinear potential theory.
This manuscript is devoted to the long-term regularity of the 2-d Navier-Stokes-Poisson system. We allow the initial density to be close to a constant and the potential part of the initial velocity to be small independently of the rescaled viscosity parameter $\varepsilon$ while the rotational part of the initial velocity is assumed to be small compared to $\varepsilon$. We then show that the lifespan of the system $T^{\varepsilon}$ satisfies $T^{\varepsilon}>\varepsilon^{-(1-\vartheta)}$, where the small constant $\vartheta$ is the size of the initial perturbation in some suitable space. The …
In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). O…
In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.
This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to $p$-degenerate elliptic equations in rough media with sources in the weak Lebesgue space $L_\text{weak}^{\frac{n}{p} + ε}$. For the borderline case, $f \in L_\text{weak}^{\frac{n}{p}}$, solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results prese…
In this work we consider the primal mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is non-standard as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher and lower regularity terms. The lower regularity terms are here explicitly known and capture the solution singularities. The higher regularity terms, meanwhile,…
We prove that the one-dimensional Euler-Poisson system driven by the Poisson forcing together with the usual γ-law pressure, γ ≥ 1, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the 2x2 p-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.
This work is concerned with the broad question of propagation of regularity for smooth solutions to non-linear Vlasov equations. For a class of equations (that includes Vlasov-Poisson and relativistic Vlasov-Maxwell), we prove that higher regularity in space is propagated, locally in time, into higher regularity for the moments in velocity of the solution. This in turn can be translated into some anisotropic Sobolev higher regularity for the solution itself, which can be interpreted as a kind of weak propagation of space regularity. To this end, we adapt the methods introduced in the context o…
We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-Δu = Q \; \mathcal{H}^{n-1} \llcorner Γ$ in a domain $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $Γ\subset \subset Ω$ is a compact (Lipschitz) manifold and $Q \in L^\infty(Γ)$. We also discuss optimality and necessity of the assumptions on $Q$ and $Γ$. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.
We show that if $Ω$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with Hölder-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $Ω$. This extends the previous work of Kenig and Toro in the case of the Laplacian.
In this paper, we study the Sobolev regularity of solutions to nonlinear second order elliptic equations with super-linear first-order terms on Riemannian manifolds, complemented with Neumann boundary conditions, when the source term of the equation belongs to a Lebesgue space, under various integrability regimes. Our method is based on an integral refinement of the Bochner's identity, and leads to "semilinear Calderón-Zygmund" type results. Applications to the problem of smoothness of solutions to Mean Field Games systems with Neumann boundary conditions posed on convex domains of the Euclide…
In this note we are concerned with interior regularity properties of the $p$-Poisson problem $Δ_p(u)=f$ with $p>2$. For all $0<λ\leq 1$ we constuct right-hand sides $f$ of differentiability $-1+λ$ such that the (Besov-) smoothness of corresponding solutions $u$ is essentially limited to $1+λ/ (p-1)$. The statements are of local nature and cover all integrability parameters. They particularly imply the optimality of a shift theorem due to Savaré [J. Funct. Anal. 152:176-201, 1998], as well as of some recent Besov regularity results of Dahlke et al. [Nonlinear Anal. 130:298-329, 2016]. Keywords:…
We consider the Cauchy problem for the barotropic Euler system coupled to Helmholtz or Poisson equations, in the whole space. We assume that the initial density is small enough, and that the initial velocity is close to some reference vector field u0 such that the spectrum of Du0 is bounded away from zero. We prove the existence of a global unique solution with (fractional) Sobolev regularity, and algebraic time decay estimates. Our work extends the papers by D. Serre and M. Grassin [20, 21, 43] dedicated to the compressible Euler system without coupling and integer regularity exponents.
In this paper, logarithmically improved regularity criteria for the Navier--Stokes/Poisson--Nernst--Planck system are established in terms of both the pressure and the gradient of pressure in the homogeneous Besov space.
We derive regularity estimates for viscosity solutions to the parabolic normalized p-Laplace. By using approximation methods and scaling arguments for the normalized p-parabolic operator, we show that the gradient of bounded viscosity solutions is locally asymptotically Lipschitz continuous when $p$ is sufficiently close to 2. In addition, we establish regularity estimates in Sobolev spaces.
Local Well-Posedness for Vlasov--Poisson with $L^{d+}$ Initial Density and Fractional Velocity Regularity
State Key Laboratory of Mathematical Sciences, Academy of Mathematics & Systems Science, The Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学学院国家重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
This paper discusses some regularity of almost periodic solutions of the Poisson's equation $-Δu = f$ in $\mathbb{R}^n$, where $f$ is an almost periodic function. It has been proved by Sibuya [Almost periodic solutions of Poisson's equation. Proc. Amer. Math. Soc., 28:195--198, 1971.] that if $u$ is a bounded continuous function and solves the Poisson's 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 s…
We study the small data global regularity problem of the $3D$ Vlasov-Poisson system for both the relativistic case and the non-relativistic case. The main goal of this paper is twofold. (i) Based on a Fourier method, which works systematically for both the relativistic case and the non-relativistic case, we give a short proof for the global regularity and the sharp decay estimate for the $3D$ Vlasov-Poisson system. Moreover, we show that the nonlinear solution scatters to a linear solution in both cases. The result of sharp decay estimates for the non-relativistic case is not new, see Hwang-Re…
We study the interrelation between the limit $L_p(Ω)$-Sobolev regularity $\overline{s}_p$ of (classes of) functions on bounded Lipschitz domains $Ω\subseteq\mathbb{R}^d$, $d\geq 2$, and the limit regularity $\overlineα_p$ within the corresponding adaptivity scale of Besov spaces $B^α_{τ,τ}(Ω)$, where $1/τ=α/d+1/p$ and $α>0$ ($p>1$ fixed). The former determines the convergence rate of uniform numerical methods, whereas the latter corresponds to the convergence rate of best $N$-term approximation. We show how additional information on the Besov or Triebel-Lizorkin regularity may be used to deduc…
This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this result is that it does not assume more than Lipschitz regularity on the boundary. Our Theorem, which is a significant improvement of our previous work on Lipschitz estimates in bumpy domains, should be read as an improved regularity result for an elliptic system over a Lipschitz boundary. Our progress in this direction is made possible by an estimate for a bounda…
In this note, we prove that if a subharmonic function $Δu\ge 0$ has pure second derivatives $\partial_{ii} u$ that are signed measures, then their negative part $(\partial_{ii} u)_-$ belongs to $L^1$ (in particular, it is not singular). We then show that this improvement of regularity cannot be upgraded to $L^p$ for any $p > 1$. We finally relate this problem to a natural question on the one-sided regularity of solutions to the obstacle problem with rough obstacles.
In this paper, we investigate the boundary Hölder regularity for elliptic equations (precisely, the Poisson equation, linear equations in divergence form and non-divergence form, the p-Laplace equations and fully nonlinear elliptic equations) on Reifenberg flat domains. We prove that for any $0<α<1$, there exists $δ>0$ such that the solution is $C^α$ at $x_0\in \partial Ω$ provided that $Ω$ is $δ$-Reifenberg flat at $x_0$ (see Definition 1.1). In particular, for any $0 < α< 1$, if $\partial Ω$ is $C^1$ and $u=g$ on $\partial Ω$ with $g\in C^α(x_0)$, then $u\in C^α(x_0)$. A similar result for t…
This article proves the regularity for the Boltzmann equation without angular cutoff with hard potential. By sharpening the coercivity and upper bound estimate on the collision operator, analyzing the Poisson bracket between the transport operator and some weighted pseudo-differential operator, we prove the regularizing effect in space and velocity variables when the initial data has mild regularity.
We study a nonlinear system made up of an elliptic equation of blended singular/degenerate type and Poisson's equation with a lowly integrable source. We prove the existence of a weak solution in any space dimension and, chiefly, derive an improved $\mathcal{C}^{1,\text{log-Lip}}$-regularity estimate using tangential analysis methods. The system illustrates a sophisticated version of the proverbial thermistor problem and our results are new even in simpler modelling scenarios.
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.
The compressible Navier-Stokes-Poisson system takes the form of usual Navier-Stokes equations coupled with the self-consistent Poisson equation, which is used to simulate the transport of charged particles under the electric field of electrostatic potential force. In this paper, we focus on the large time behavior of global strong solutions in the $L^{p}$ Besov spaces of critical regularity. By exploring the dissipative effect arising from Poisson potential, we posed the new regularity assumption of low frequencies and then establish a \textit{sharp} time-weighted inequality, which leads to th…
An $\eps$-regularity criterion at one scale for 3-D Navier-Stokes equations and improved upper bound on the box dimension of the interior singular set
Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University（河北大学数学与信息科学学院机器学习与计算智能河北省重点实验室）
We study the semilinear Poisson equation \begin{equation} \label{pro} Δu = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp.
In the present work we establish sharp regularity estimates for the solutions of the porous medium equation, along their zero level-sets. We work under a proximity regime on the exponent governing the nonlinearity of the problem. Then, we prove that solutions are locally of class $\mathcal{C}^{1-,\frac{1}{2}-}$ along free boundary points $x_0\in \partial\left\lbrace u>0\right\rbrace$, both in time and space. Our argument consists of importing information from the heat equation, through approximation and localization methods.
We show how Hölder estimates for Feller semigroups can be used to obtain regularity results for solutions to the Poisson equation $Af=g$ associated with the (extended) infinitesimal generator $A$ of a Feller process. The regularity of $f$ is described in terms of Hölder-Zygmund spaces of variable order and, moreover, we establish Schauder estimates. Since Hölder estimates for Feller semigroups have been intensively studied in the last years, our results apply to a wide class of Feller processes, e.g. random time changes of Lévy processes and solutions to Lévy-driven stochastic differential equ…
In this note we prove an abstract version of a recent quantitative stratifcation priciple introduced by Cheeger and Naber (Invent. Math., 191 (2013), no. 2, 321-339; Comm. Pure Appl. Math., 66 (2013), no. 6, 965-990). Using this general regularity result paired with an $\varepsilon$-regularity theorem we provide a new estimate of the Minkowski dimension of the set of higher multiplicity points of a Dir-minimizing Q-valued function. The abstract priciple is applicable to several other problems: we recover recent results in the literature and we obtain also some improvements in more classical co…
