Related papers: On the regularity of continuous solutions to multi…
In this paper we shall establish some regularity results of solutions of a class of fully nonlinear equations, with a first order term which is sub-linear. We prove local H\"older regularity of the gradient both in the interior and up to…
This is a note on \cite{LSU} and \cite{FS}. Using their work line by line, we prove the H\"older-continuity of solutions to linear parabolic equations of mixed type, assuming the coefficient of $\frac{\partial}{\partial t}$ has…
In this article, we present a method to find a solution to a one-dimensional nonlocal conservation law that respects a space-dependent mapping, referred to as the obstacle. This is achieved by generalizing existing results for the local…
We consider small nonlinear perturbations of linear systems on a time scale with the phase space being finite or infinite-dimensional. For $\Delta$-differential operators, corresponding to linear dynamic systems we consider their…
Following Dibenedetto's intrinsic scaling method, we prove local H\"older continuity of weak solutions to obstacle problems related to some anisotropic parabolic equations under the condition for which only H\"older's continuity of the…
In this paper we study the regularity property of Hele-Shaw flow, where source and drift are present in the evolution. More specifically we consider H\"{o}lder continuous source and Lipschitz continuous drift. We show that if the free…
We show uniqueness and stability in $L^2$ and for all time for piecewise-smooth solutions to hyperbolic balance laws. We have in mind applications to gas dynamics, the isentropic Euler system and the full Euler system for a polytropic gas…
Stochastic non-local conservation law equation in the presence of discontinuous flux functions is considered in an $L^{1}\cap L^{2}$ setting. The flux function is assumed bounded and integrable (spatial variable). Our result is to prove…
We prove the existence of BV solutions for $2\times 2$ system of hyperbolic balance laws in one space dimension. The flux is assumed to have two genuinely nonlinear characteristic fields. We consider a general force which may possibly…
Let $u(t,x)$ be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution…
We prove Holder continuity for solutions to the n-dimensional H-System assuming logarithmic higher integrability of the solution.
We consider the H\"older regularity of solutions to the steady Boltzmann equation with in-flow boundary condition in bounded and strictly convex domains $\Omega\subset\mathbb{R}^{3}$ for gases with cutoff soft potential $(-3<\gamma<0)$. We…
We consider a class of abstract quasilinear parabolic problems with lower--order terms exhibiting a prescribed singular structure. We prove well--posedness and Lipschitz continuity of associated semiflows. Moreover, we investigate global…
We consider a general family of nonlocal in space and time diffusion equations with space-time dependent diffusivity and prove convergence of finite difference schemes in the context of viscosity solutions under very mild conditions. The…
We obtain several new regularity results for solutions of scalar conservation laws satisfying the genuine nonlinearity condition. We prove that the solutions are continuous outside of the jump set, which is codimension one rectifiable. We…
We consider nxn hyperbolic systems of balance laws in one-space dimension under the assumption that all negative (resp. positive) characteristics are linearly degenerate. We prove the local exact one-sided boundary null controllability of…
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 are interested in viscous scalar conservation laws with a white-in-time but spatially correlated stochastic forcing. The equation is assumed to be one-dimensional and periodic in the space variable, and its flux function to be locally…
We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely…
In this paper, we show that the entropy solution of a scalar conservation law is - continuous outside a $1$-rectifiable set $\Xi$, - up to a $\mathcal H^1$ negligible set, for each point $(\bar t,\bar x) \in \Xi$ there exists two regions…