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We consider the 2-D incompressible Euler equations in a bounded domain and show that local weak solutions are exponentially integrable, uniformly in time, under minimal integrability conditions. This is a Serrin-type interior regularity…
In the density estimation model, we investigate the problem of constructing adaptive honest confidence sets with radius measured in Wasserstein distance $W_p$, $p\geq1$, and for densities with unknown regularity measured on a Besov scale.…
We consider the Dirichlet problem for elliptic systems with periodically distributed inclusions whose conduction parameter exhibits a significant contrast compared to the background media. We develop a unified method to quantify the…
It is known that linear advection equations with Sobolev velocity fields have very poor regularity properties: Solutions propagate only derivatives of logarithmic order, which can be measured in terms of suitable Gagliardo seminorms. We…
This paper is concerned with boundary regularity estimates in the homogenization of elliptic equations with rapidly oscillating and high-contrast coefficients. We establish uniform nontangential-maximal-function estimates for the Dirichlet,…
This work investigates finite element approximations for a general class of elliptic hemivariational inequalities arising in semipermeable media. The proposed model incorporates non-isotropic and heterogeneous diffusion coefficients,…
In this paper, we establish the interior Hessian estimates for $2$-convex solutions to $\frac{\sigma_2}{\sigma_1} (D^2 u) = \psi (x,u)$ in dimension three. In higher dimensions ($n \geq 4$), we prove the interior Hessian estimates for…
We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy,…
We consider regularised quadratic optimal transport with subquadratic polynomial or entropic regularisation. In both cases, we prove interior Lipschitz-estimates on a transport-like map and interior gradient Lipschitz-estimates on the…
In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first order…
We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla…
Due to the highly degeneracy and singularities of the entropy equation, the physical entropy for viscous and heat conductive polytropic gases behave singularly in the presence of vacuum and it is thus a challenge to study its dynamics. It…
We establish a priori regularity estimates for viscosity solutions of degenerate fully nonlinear elliptic equations with integrable right-hand sides. When the nonhomogeneous term belongs to $L^p$ with $p>n$, we prove optimal interior…
This paper is concerned with the inhomogeneous incompressible Euler system. We establish a Duchon--Robert type approximation theorem for the distribution describing the local energy flux of bounded solutions. The velocity field is assumed…
In this paper, we consider stochastic homogenization of elliptic equations with unbounded and non-uniformly elliptic coefficients. Extending subadditive arguments, we get an estimate for the rate of the convergence of the solution of the…
We consider the regularity of stationary solutions to the linearized Boltzmann equations in bounded $C^1$ convex domains in $\mathbb{R}^3$ for gases with cutoff hard potential and cutoff Maxwellian gases. We prove that the stationary…
We consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with an extreme conductivity (insulators or perfect conductors) are closely located, the…
We investigate the regularity of the viscosity solutions to a class of degenerate/singular fully nonlinear elliptic equations with Hamiltonian terms. To overcome the difficulty caused by the simultaneous presence of the general…
This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a nonlinear, quantitative version of a growth lemma established by…
The partial differential equations describing compressible fluid flows can be notoriously difficult to resolve on a pragmatic scale and often require the use of high performance computing systems and/or accelerators. However, these systems…