Weak-strong Uniqueness Result Research Articles

Diffuse-interface approximation and weak–strong uniqueness of anisotropic mean curvature flow

Abstract The purpose of this paper is to derive anisotropic mean curvature flow as the limit of the anisotropic Allen–Cahn equation. We rely on distributional solution concepts for both the diffuse and sharp interface models and prove convergence using relative entropy methods, which have recently proven to be a powerful tool in interface evolution problems. With the same relative entropy, we prove a weak–strong uniqueness result, which relies on the construction of gradient flow calibrations for our anisotropic energy functionals.

Relative energy inequality and weak-strong uniqueness for an isothermal non-Newtonian compressible fluid

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a de­ri­va­tion of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the strong solution exists. For this purpose, a new assumption on the coercivity of the viscous stress tensor was introduced along with two natural examples satisfying it.

Weak-strong uniqueness and vanishing viscosity for incompressible Euler equations in exponential spaces

In the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to Lloc1([0,+∞);Lexp(Rd;Rd×d)), where Lexp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray–Hopf weak solutions of the Navier–Stokes equations.

Weak–strong uniqueness for the isentropic Euler equations with possible vacuum

We establish a weak–strong uniqueness result for the isentropic compressible Euler equations, that is: As long as a sufficiently regular solution exists, all energy-admissible weak solutions with the same initial data coincide with it. The main novelty in this contribution, compared to previous literature, is that we allow for possible vacuum in the strong solution.

Weak-strong uniqueness for measure-valued solutions to the equations of quasiconvex adiabatic thermoelasticity

This article studies the equations of adiabatic thermoelasticity endowed with an internal energy satisfying an appropriate quasiconvexity assumption which is associated to the symmetrisability condition for the system. A Gårding-type inequality for these quasiconvex functions is proved and used to establish a weak-strong uniqueness result for a class of dissipative measure-valued solutions.

Existence, regularity and weak-strong uniqueness for three-dimensional Peterlin viscoelastic model

In this paper we analyze the three-dimensional Peterlin viscoelastic model. By means of a mixed Galerkin and semigroup approach we prove the existence of a weak solutions. Further combining parabolic regularity with the relative energy method we derive a conditional weak-strong uniqueness result.

A Uniqueness Result for 3D Incompressible Fluid-Rigid Body Interaction Problem

We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy the Prodi-Serrin $$\text {L}^{r}-\text {L}^{s}$$ condition are unique in the class of Leray-Hopf weak solutions.

Global well-posedness of strong solutions with large oscillations and vacuum to the compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile

This paper concerns an initial value problem of compressible Navier-Stokes-Poisson equations subject to large and non-flat doping profile in whole space R3. The global well-posedness of strong solutions with large oscillations and vacuum is established, provided the initial data are of small energy and the steady state is strictly away from vacuum. A weak-strong uniqueness result is also obtained.

English

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 weak-strong uniqueness result for this system in a bounded domain in three spatial dimensions which implies that when a strong solution exists, then a weak solution emanating from the same data coincides with the strong solution on its whole life span. The proof of given assertion relies on a form of a relative entropy method.

Markov Selections for the Magnetohydrodynamics and the Hall-Magnetohydrodynamics Systems

The magnetohydrodynamics system has found rich applications in applied sciences and has history of intense investigation by researchers including mathematicians. On the other hand, the Hall-magnetohydrodynamics system consists of the former system added by a Hall term which is very singular, and the mathematical development on this system has seen its significant progress only in the last several years. In short, the magnetohydrodynamics system, similarly to the Navier–Stokes equations, is semilinear, while the Hall-magnetohydrodynamics system is quasilinear. In this manuscript, we consider the three-dimensional magnetohydrodynamics, as well as the Hall-magnetohydrodynamics systems, and prove that they have Markov selections. In the case of the magnetohydrodynamics system, we prove furthermore a weak–strong uniqueness result and that any Markov solution has the strong Feller property for regular initial conditions. Consequently, it is deduced that if the magnetohydrodynamics system is well posed starting from one initial data, then it is well posed starting from any initial data, verifying a sharp contrast to the deterministic case in which the well posedness for all time with small initial data is well known. In the case of the Hall-magnetohydrodynamics system, we are also able to prove the weak–strong uniqueness result; however, the proof of strong Feller property seems to break down due to the singularity of the Hall term, about which we elaborate in detail.

Weak-strong uniqueness for fluid-rigid body interaction problem with slip boundary condition

We consider a coupled partial differential equation-ordinary differential equation system describing the motion of the rigid body in a container filled with the incompressible, viscous fluid. The fluid and the rigid body are coupled via Navier’s slip boundary condition. We prove that the local in time strong solution is unique in the larger class of weak solutions on the interval of its existence. This is the first weak-strong uniqueness result in the area of fluid-structure interaction with a moving boundary.

Uniqueness of solutions for a mathematical model for magneto-viscoelastic flows

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 gradient and for the magnetization obtained from a special case of the micromagnetic energy. It turns out that the conditions on uniqueness coincide with those for the well-known Navier–Stokes equations in bounded domains: weak solutions are unique in two spatial dimensions, and weak solutions satisfying the Prodi–Serrin conditions are unique among all weak solutions in three dimensions. That is, we obtain the so-called weak-strong uniqueness result in three spatial dimensions.

A symmetrizable extension of polyconvex thermoelasticity and applications to zero-viscosity limits and weak-strong uniqueness

ABSTRACTWe embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system and derive a relative entropy identity in the extended variables. Following the relative entropy formulation, we prove the convergence from thermoviscoelasticity with Newtonian viscosity and Fourier heat conduction to smooth solutions of the system of adiabatic thermoelasticity as both parameters tend to zero. Also, convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. Finally, we establish a weak-strong uniqueness result for the equations of adiabatic thermoelasticity in the class of entropy weak solutions.

The semi-geostrophic system: weak-strong uniqueness under uniform convexity

We prove a weak-strong uniqueness result for the semi-geostrophic system with constant Coriolis force. The main assumptions on the strong solution are the boundedness of the velocity field as well as the uniform convexity of the Legendre-Fenchel transform of the modified pressure. We give several examples where our results apply, including some classical solutions on the 2-dimensional torus, and the “stationary” solutions for 3DSG (for which the total wind velocity is zero but the pressure may be time-dependent).

Relative entropy and a weak–strong uniqueness principle for the compressible Navier–Stokes equations on moving domains

We establish a relative entropy inequality for the compressible Navier–Stokes equations, posed on domains with time-dependent moving boundaries. Using the relative entropy, a weak–strong uniqueness result is shown for the class of finite energy weak solutions.

Weakly dissipative solutions and weak–strong uniqueness for the Navier–Stokes–Smoluchowski system

This article deals with a fluid–particle interaction model for the evolution of particles dispersed in a fluid. The fluid flow is governed by the Navier–Stokes equations for a compressible fluid while the evolution of the particle densities is given by the Smoluchowski equation. The coupling between the dispersed and dense phases is obtained through the drag forces that the fluid and the particles exert mutually. The existence of weakly dissipative solutions is established under reasonable physical assumptions on the initial data, the physical domain, and the external potential. Furthermore, a weak–strong uniqueness result is established via the relative entropy method yielding that a weakly dissipative solution agrees with a classical solution with the same initial data when such a classical solution exists.

Initial Boundary Value Problem of the General Three-Component Camassa-Holm Shallow Water System on an Interval

We study the initial boundary value problem of the general three-component Camassa-Holm shallow water system on an interval subject to inhomogeneous boundary conditions. First we prove a local in time existence theorem and present a weak-strong uniqueness result. Then, we establish a asymptotic stabilization of this system by a boundary feedback. Finally, we obtain a result of blow-up solution with certain initial data and boundary profiles.

Weak–strong uniqueness for the generalized Navier–Stokes equations

We give a weak–strong uniqueness result for the weak solutions of the generalized Navier–Stokes equations in Besov space.

Initial boundary value problem and asymptotic stabilization of the Camassa–Holm equation on an interval

We investigate the nonhomogeneous initial boundary value problem for the Camassa–Holm equation on an interval. We provide a local in time existence theorem and a weak-strong uniqueness result. Next we establish a result on the global asymptotic stabilization problem by means of a boundary feedback law.