Related papers: Strong solutions and weak-strong stability in a sy…
In this work, we introduce a notion of dissipative weak solution for a system describing the evolution of a heat-conducting incompressible non-Newtonian fluid. This concept of solution is based on the balance of entropy instead of the…
We study the existence and the rate of equilibration of weak solutions to a two-component system of non-linear diffusion-aggregation equations, with small cross diffusion effects. The aggregation term is assumed to be purely attractive, and…
In this work we investigate the existence of weak solutions for steady flows of generalized incompressible and homogeneous viscous fluids. The problem is modeled by the steady case of the generalized Navier-Stokes equations, where the…
We establish a weak-strong uniqueness principle for the two-phase Mullins-Sekerka equation in the plane: As long as a classical solution to the evolution problem exists, any weak De Giorgi type varifold solution (see for this notion the…
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…
The steady problem resulting from a mixture of two distinct fluids of power-law type is analyzed in this work. Mathematically, the problem results from the superposition of two power laws, one for a constant power-law index with other for a…
We establish stability properties of weak solutions for systems of porous medium type with respect to the exponent $m$. Thereby we treat stability for the local case as well as for Cauchy-Dirichlet problems. Both degenerate and singular…
We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation…
We discuss the analysis and stability of a family of cross-diffusion boundary value problems with nonlinear diffusion and drift terms. We assume that these systems are close, in a suitable sense, to a set of decoupled and linear problems.…
We study a one-dimensional cross-diffusion system for two interacting populations on the torus, with a fast-diffusion law with exponent $0< \alpha\le 1$ and different external potentials. For arbitrary non-negative $L^{1}$ initial data with…
The Navier-Stokes-Fourier system describing the motion of a compressible, viscous, and heat conducting fluid is known to possess global-in-time weak solutions for any initial data of finite energy. We show that a weak solution coincides…
We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including…
Variations of loading level and changes in system topological property may cause the operating point of an electric power systems to move gradually towards the verge of its transmission capability, which can lead to catastrophic outcomes…
First, a new sufficient condition for uniqueness of weak solutions is proved for the system of 2D viscous Primitive Equations. Second, global existence and uniqueness are established for several classes of weak solutions with partial…
The coupled Navier-Stokes/Allen-Cahn system is a simple model to describe phase separation in two-component systems interacting with an incompressible fluid flow. We demonstrate the \emph{weak-strong uniqueness} result for this system in a…
We consider the Cauchy problem for incompressible viscoelastic fluids in the whole space $\mathbb{R}^d$ ($d=2,3$). By introducing a new decomposition via Helmholtz's projections, we first provide an alternative proof on the existence of…
So far existence of dissipative weak solutions for the compressible Navier-Stokes equations (i.e. weak solutions satisfying the relative energy inequality) is known only in the case of boundary conditions with non zero inflow/outflow (i.e.,…
Following Abbatiello et al. [ DCCDS-A (41), 2020], we introduce dissipative turbulent solutions to a simple model of a mixture of two non interacting compressible fluids {\tc filling a bounded domain with general non zero inflow/outflow…
We consider a system of evolutionary equations that is capable of describing certain viscoelastic effects in linearized yet nonlinear models of solid mechanics. The essence of the paper is that the constitutive relation, involving the…
We study the (weak) equilibrium problem arising from the problem of optimally stopping a one-dimensional diffusion subject to an expectation constraint on the time until stopping. The weak equilibrium problem is realized with a set of…