Related papers: Lipschitz continuity of solutions to drift-diffusi…

In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain informations on the pressure. This fact has been…

We show that locally bounded, local weak solutions to certain nonlocal, nonlinear diffusion equations modeled on the fractional porous media and fast diffusion equations given by \begin{align*} \partial_t u + (-\Delta)^s(|u|^{m-1}u) = 0…

We address the long-time behavior of the 2D Boussinesq system, which consists of the incompressible Navier-Stokes equations driven by a non-diffusive density. We construct globally persistent solutions on a smooth bounded domain, when the…

Here we prove the all-time propagation of the Sobolev regularity for the velocity field solution of the two-dimensional compressible Navier-Stokes equations, provided the volume (bulk) viscosity coefficient is large enough. The initial…

We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the…

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous…

For periodic initial data with the density allowing vacuum, we establish the global existence and exponential decay of weak, strong and classical solutions to the two-dimensional(2D) compressible Navier-Stokes equations when the bulk…

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite…

We study the regularity of the bounded self-similar solution to the one-phase Stefan problem with fractional diffusion posed on the whole line. In terms of the enthalpy $h(x,t)$, the evolution problem reads \[ \begin{cases} \partial_t h +…

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier-Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space $\dot H^{1/2}$…

We consider solutions of the Navier-Stokes equation with fractional dissipation of order $\alpha\geq 1$. We show that for any divergence-free initial datum $u_0$ such that $||u_0||_{H^{\delta}} \leq M$, where $M$ is arbitrarily large and…

We investigate regularity and a priori estimates for Fokker-Planck and Hamilton-Jacobi equations with unbounded ingredients driven by the fractional Laplacian of order $s\in(1/2,1)$. As for Fokker-Planck equations, we establish…

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…

In this book we establish under suitable assumptions the uniqueness and existence of viscosity solutions of Kolmogorov backward equations for stochastic partial differential equations (SPDEs). In addition, we show that this solution is the…

In this paper, we analyze the Nitsche's method for the stationary Navier-Stokes equations on Lipschitz domains under minimal regularity assumptions. Our analysis provides a robust formulation for implementing slip (i.e. Navier) boundary…

In their classical work [20], Caflisch and Sammartino established the inviscid limit and boundary layer expansions of vanishing viscosity solutions to the incompressible Navier-Stokes equations for analytic data on a half-space. It was then…

The study of flows over an obstacle is one of the fundamental problems in fluids. In this paper we establish the global validity of the diffusive limit for the Boltzmann equations to the Navier-Stokes-Fourier system in an exterior domain.…

Motivated by the critical dissipative quasi-geostrophic equation, we prove that drift-diffusion equations with L^2 initial data and minimal assumptions on the drift are locally Holder continuous. As an application we show that solutions of…

We establish interior Lipschitz regularity for continuous viscosity solutions of fully nonlinear, conformally invariant, degenerate elliptic equations. As a by-product of our method, we also prove a weak form of the strong comparison…

We study the existence of global weak solutions of a nonlinear transport-diffusion equation with a fractional derivative in the time variable and under some extra hypotheses, we also study some regularity properties for this type of…