相关论文: On the $L^p$ regularity of solutions to the genera…

The Swift-Hohenberg equation is ubiquitous in the study of bistable dynamics. In this paper, we study the dynamic transitions of the Swift-Hohenberg equation with a third-order dispersion term in one spacial dimension with a periodic…

We consider integrability structures of the generalized Hunter--Saxton equation. In particular, we obtain the Lax representation with nonremovable spectral parameter, find local recursion operators for symmetries and cosymmetries, generate…

We study existence and regularity properties of solutions to the singular $p$-Laplacean parabolic system in a bounded domain $\Omega$. The main purpose is to prove global $L^r(\varepsilon,T;L^q(\Omega))$, $\varepsilon\geq0$, integrability…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce…

We present a general blow-up technique to obtain local regularity estimates for solutions, and their derivatives, of second order elliptic equations in divergence form in H\"older spaces with variable exponent. The procedure allows to…

We are concerned with existence of regular solutions for non-Newtonian fluids in dimension three. For a certain type of non-Newtonian fluids we prove local existence of unique regular solutions, provided that the initial data are…

The main purpose of this work is to study uniform regularity estimates for a family of elliptic operators $\{\mathcal{L}_\varepsilon, \varepsilon>0\}$, arising in the theory of homogenization, with rapidly oscillating periodic coefficients.…

A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical…

A chemotaxis system possibly containing rotational components of the cross-diffusive flux is studied under no-flux boundary conditions in a bounded domain $\Omega\subset R^n$, $n\ge 1$, with smooth boundary, where the evolution of the…

We investigate a fluid-structure interaction system in which the dynamics of the fluid is described by the compressible Navier-Stokes equations, while the elastic structure is modeled by a damped plate equation. The fluid evolves in a…

We consider nonlocal initial boundary value problems with integral boundary conditions for integro-differential first order hyperbolic systems. We prove a general regularity result stating that the $L^2$-generalized solutions become…

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou \cite{HouLuo14} proposed a new finite time blow up scenario based on extensive numerical…

The $L^{p}$ theory for non-isentropic Navier-Stokes equations governing compressible viscous and heat-conductive gases is not yet proved completely so far, because the critical regularity cannot control all non linear coupling terms. In…

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first…

Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh. This system is Galilean invariant, linearly non-dispersive and conserves formally an $H^1$-like energy. In this paper, we…

We consider an abstract second order linear equation with a strong dissipation, namely a friction term which depends on a power of the "elastic" operator. In the homogeneous case, we investigate the phase spaces in which the initial value…

This paper is devoted to the study of the regularity of solutions to some systems of reaction--diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without…

We study an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Gr\"{u}n in 2012. This model arises in the diffuse interface theory for binary…

In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel…

We study free boundary compressible viscous models that may include nonlinear viscosities. These are compressible/incompressible Navier-Stokes type systems for a non-Newtonian stress tensor. They describe the motion of a possibly…