Related papers: Parabolic equations with rough data
We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic $p$-Laplace equation. We show that, up to a…
We present a local almost everywhere regularity result for a general nonlinear non-diagonal parabolic system, which main part depends on symmetric part of the gradient.
In this paper we derive some a priori estimates for a class of linear coagulation equations with particle fluxes towards large size particles. The derived estimates allow us to prove local well posedness for the considered equations. Some…
We prove the local boundedness of local weak solutions to the parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…
There is a tendency to write the equations of general relativity as a first order symmetric system of time dependent partial differential equations. However, for numerical reasons, it might be advantageous to use a second order formulation…
This article provides a survey on some main results and recent developments in the mathematical theory of water waves. More precisely, we briefly discuss the mathematical modeling of water waves and then we give an overview of local and…
In this note, we review some of the recent developments in the well-posedness theory of nonlinear dispersive partial differential equations with random initial data.
In this paper, we study linear backward parabolic SPDEs in bounded domains and present new a priori estimates for their weak solutions. Inspired by the seminal work of Y. Hu, J. Ma and J. Yong from 2002 on strong solutions, we establish…
We investigate the existence of a renormalized solution for a class of nonlinear parabolic equations with two lower order terms and $L^1$-data.
We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local…
We establish the global existence of a class of strongly coupled parabolic systems. The necessary apriori estimates will be obtained via our new approach to the regularity theory of parabolic scalar equations with integrable data and new…
We establish a new type of weak Harnack estimates with optimal parabolic tail for the weak supersolutions to a doubly nonlinear nonlocal $p$-Laplace equation, which is modeled on the nonlocal Trudinger equation. Our results are achieved by…
We prove local in time well-posedness for a large class of quasilinear Hamiltonian, or parity preserving, Schr\"odinger equations on the circle. After a paralinearization of the equation, we perform several paradifferential changes of…
An averaging method for getting uniformly valid asymptotic approximations of the solution of hyperbolic systems of equations is presented. The averaged system of equations disintegrates into independent equations for non-resonance systems.…
We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on…
Suppose an interval is put on a horizontal line with random roughness. With probability one it is supported at two points, one from the left, and another from the right from its center. We compute probability distribution of support points…
In this work we investigate the origin of the parabolic relation between skewness and kurtosis often encountered in the analysis of experimental time-series. We argue that the numerical values of the coefficients of the curve may provide…
We address the local well-posedness of the hydrostatic Navier-Stokes equations. These equations, sometimes called reduced Navier-Stokes/Prandtl, appear as a formal limit of the Navier-Stokes system in thin domains, under certain constraints…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…
We revisit the local well-posedness theory of nonlinear Schr\"odinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data…