Related papers: On the Boundary Regularity of Weak Solutions to th…
We prove partial and full boundary regularity for manifold constrained $p(x)$-harmonic maps.
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…
This is an expository paper on the theory of local regularity for weak solutions to the non-stationary 3D Navier-Stokes equations near the boundary of a domain.
In this paper we first study partial regularity of weak solutions to the initial boundary value problem for the system $-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \ \partial_t\mathbf{m}-D^2\Delta…
We study the weak solutions to the electron-MHD system and obtain a conditional uniqueness result. In addition, we prove conservation of helicity for weak solutions to the electron-MHD system under a geometric condition.
In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.
We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here…
We consider a boundary value problem driven by the fractional p-Laplacian operator with a bounded reaction term. By means of barrier arguments, we prove H\"older regularity up to the boundary for the weak solutions, both in the singular…
This paper is concerned with a compressible MHD equations describing the evolution of viscous non-resistive fluids in piecewise regular bounded Lipschitz domains. Under the general inflow-outflow boundary conditions, we prove existence of…
We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of…
In this paper we investigate regularity properties of weak solutions to a PDE system that arises in the study of biological transport networks. The system consists of a possibly singular elliptic equation for the scalar pressure of the…
In this paper we study on smooth bounded domains the global regularity (up to the boundary) for weak solutions to systems having $p$-structure depending only on the symmetric part of the gradient.
The magnetohydrodynamics (MHD) problem is most often studied in a framework where Dirichlet type boundary conditions on the velocity field is imposed. In this Note, we study the (MHD) system with pressure boundary condition, together with…
In this paper, we study the regularity of several notions of Lipschitz solutions to the minimal surface system with an emphasis on partial regularity results. These include stationary solutions, integral weak solutions, and viscosity…
To verify theoretical results it is sometimes important to use a numerical example where the solution has a particular regularity. The paper describes one approach to construct such examples. It is based on the regularity theory for…
We study the regularity properties of a weak solution to the boundary value problem for the equation $-\Delta \rho +a u=f$ in a bounded domain $\Omega\subset \mathbb{R}^N$, where $\rho=e^{-\mbox{div}\left(|\nabla u|^{p-2}\nabla…
We show that finite-energy weak solutions to the incompressible Navier--Stokes equations on a three-dimensional bounded smooth domain are regular up to the boundary, provided that the $L^4_tL^4_x$-norm of the solution is smaller than a…
We establish a new regularity property for weak solutions of parabolic systems with coefficients depending measurably on time as well as on all spatial variables. Namely, weak solutions are locally H{\"o}lder continuous Lp valued functions…
We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions…
We establish the pointwise continuity of bounded weak solutions to of a class of scalar parabolic equations and strongly coupled parabolic systems. Our approach to the regularity theory of parabolic scalar equations is quite elementary and…