Related papers: Partial regularity for parabolic systems with VMO-…

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of…

We consider nonlinear Kolmogorov-Fokker-Planck type equations of the form \begin{equation}\label{abeqn} (\partial_t+X\cdot\nabla_Y)u=\nabla_X\cdot(A(\nabla_X u,X,Y,t)). \end{equation} The function…

We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the…

We revisit the partial $\mathrm{C}^{1,\alpha}$ regularity theory for minimizers of non-parametric integrals with emphasis on sharp dependence of the H\"older exponent $\alpha$ on structural assumptions for general zero-order terms. A…

We examine the conditional regularity of the solutions of Navier-Stokes equations in the entire three-dimensional space under the assumption that the data are axially symmetric. We show that if positive part of the radial component of…

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the…

We consider the problem of spectral stability of traveling wave solutions $u=\gamma(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories…

For proper minimizers of parabolic variational integrals with linear growth with respect to $|Du|$, we establish a necessary and sufficient condition for $u$ to be continuous at a point $(x_o,t_o)$, in terms of a sufficient fast decay of…

We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher…

The main result of this paper is to prove that viscosity solutions to a parabolic free boundary problem with variable coefficients are Lipschitz continuous under the assumptions that the solution has a Lipschitz free boundary and satisfies…

In this paper, we consider higher regularity of a weak solution $({\bf u},p)$ to stationary Stokes systems with variable coefficients. Under the assumptions that coefficients and data are piecewise $C^{s,\delta}$ in a bounded domain…

We prove that the moduli of continuity of viscosity solutions to fully nonlinear parabolic partial differential equations are viscosity subsolutions of suitable parabolic equations of one space variable. As applications, we obtain sharp…

In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in $\mathbb{R}^{n}$ for $n=2,\,3$. In comparison with the work of the 3D fractional Navier-Stokes equations obtained…

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only…

We address the quantitative uniqueness properties of the solutions of the parabolic equation $ \partial_t u - \Delta u = w_j (x,t) \partial_j u + v(x,t) u $ where $v$ and $w$ are bounded. We prove that for solutions $u$, the order of…

We consider the incompressible Navier-Stokes equations in a moving domain whose boundary is prescribed by a function $\eta=\eta(t,y)$ (with $y\in\mathbb R^2$) of low regularity. This is motivated by problems from fluid-structure…

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted…

We consider solutions of two-dimensional $m \times m$ systems hyperbolic conservation laws that are constant in time and along rays starting at the origin. The solutions are assumed to be small $L^\infty$ perturbations of a constant state…

Based on a compactness method, we establish regularity criteria for suitable weak solutions to the surface growth model with a forcing term. These criteria imply that the H\"older regularity of solutions follows from smallness conditions on…

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1\Delta u+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2\Delta v+u^q\quad & \quad\mbox{in}\quad{\bf…