Related papers: Regularity of solutions to the Muskat equation
In this article we study the existence and uniqueness of solutions of stochastic continuity equation with irregular coefficients.
Regularity and uniqueness of weak solution of the compressible isentropic Navier-Stokes equations is proven for small time in dimension $N=2,3$ under periodic boundary conditions. In this paper, the initial density is not required to have a…
The Monge-Amp\`ere type equations over bounded convex domains arise in a host of geometric applications. In this paper, we focus on the Dirichlet problem for a class of Monge-Amp\`ere type equations, which can be degenerate or singular near…
The mixing problem is classically encountered in the study of differential equations applied to fluid dynamics. An understanding of fluid movement under constraints is particularly important in the field of medicine as many therapeutics and…
The problem of describing the behavior of the solutions to the Navier-Stokes equations in three space dimensions has always been borderline. From one side, due to the viscosity term, smooth data seem to produce solutions with an everlasting…
A Stokes-type problem for a viscoelastic model of salt rocks is considered, and existence, uniqueness and regularity are investigated in the scale of $L^2$-based Sobolev spaces. The system is transformed into a generalized Stokes problem,…
In this paper we prove the existence of nonsmooth viscosity solutions for Dirichlet problems involving elementary symmetric functions of the eigenvalues of the complex Hessian.
A fundamental open problem in the theory of the multidimensional compressible Navier-Stokes equations is whether smooth solutions can develop singularities in finite time. For constant viscosity coefficients, recent remarkable results show…
It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup…
It is shown that the one-dimensional or two-dimensional radially symmetric isothermal compressible Navier-Stokes system has no non-trivial global smooth solutions if the initial density is compactly supported. This result is a…
In order to find a better physical model to describe the large-scale cloud-water transformation and rainfall, we consider a moist atmosphere model consisting of the primitive equations with only horizontal viscosity in the dynamic equation…
We study a class of second order variational inequalities with bilateral constraints. Under certain conditions we show the existence of a unique viscosity solution of these variational inequalities and give a stochastic representation to…
The Assouad and lower dimensions and dimension spectra quantify the regularity of a measure by considering the relative measure of concentric balls. On the other hand, one can quantify the smoothness of an absolutely continuous measure by…
Einstein equations are addressed with the energy-momentum tensor that appears if the equations under discussion are required to possess conformal invariance. It is proved that thus derived equations (equations of conformally invariant…
In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type…
A classical model of fluid dynamics is considered which describes the shape evolution of a viscous liquid droplet on a homogeneous substrate. All equilibria are characterized and their stability is analyzed by a geometric reduction…
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular…
We discuss several approaches to generalized solutions of problems describing the motion of inviscid fluids. We propose a new concept of dissipative solution to the compressible Euler system based on a careful analysis of possible…
We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids…
In this paper we first obtain the existence of smooth solutions to Orlicz-Aleksandrov problem via a Gauss-like curvature flow.