Relative Energy Inequality Research Articles

Global Existence and Weak-Strong Uniqueness for Chemotaxis Compressible Navier–Stokes Equations Modeling Vascular Network Formation

A model of vascular network formation is analyzed in a bounded domain, consisting of the compressible Navier–Stokes equations for the density of the endothelial cells and their velocity, coupled to a reaction-diffusion equation for the concentration of the chemoattractant, which triggers the migration of the endothelial cells and the blood vessel formation. The coupling of the equations is realized by the chemotaxis force in the momentum balance equation. The global existence of finite energy weak solutions is shown for adiabatic pressure coefficients γ>8/5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma >8/5$$\\end{document}. The solutions satisfy a relative energy inequality, which allows for the proof of the weak–strong uniqueness property.

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.

Incompressible limit for a fluid mixture

In this paper we discuss the incompressible limit for multicomponent fluids in the isothermal ideal case. Both a direct limit-passage in the equation of state and the low Mach-number limit in rescaled PDEs are investigated. Using the relative energy inequality, we obtain convergence results for the densities and the velocity-field under the condition that the incompressible model possesses a sufficiently smooth solution, which is granted at least for a short time. Moreover, in comparison to single-component flows, uniform estimates and the convergence of the pressure are needed in the multicomponent case because the incompressible velocity field is not divergence-free. We show that certain constellations of the mobility tensor allow to control gradients of the entropic variables and yield the convergence of the pressure in L1.

Improved error estimates for the finite volume and the MAC schemes for the compressible Navier–Stokes system

We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier–Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.

Compressible Fluids Interacting with Plates: Regularity and Weak-Strong Uniqueness

In this paper, we study a nonlinear interaction problem between compressible viscous fluids and plates. For this problem, we introduce relative entropy and relative energy inequality for the finite energy weak solutions (FEWS). First, we prove that for all FEWS, the structure displacement enjoys improved regularity by utilizing the dissipation effects of the fluid onto the structure and that all FEWS satisfy the relative energy inequality. Then, we show that all FEWS enjoy the weak-strong uniqueness property, thus extending the classical result for compressible Navier–Stokes system to this fluid-structure interaction problem.

Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models

Abstract We study a model for a fluid showing viscoelastic and viscoplastic behavior, which describes the flow in terms of the fluid velocity and a symmetric deviatoric stress tensor. This stress tensor is transported via the Zaremba-Jaumann rate, and it is subject to two dissipation processes: one induced by a nonsmooth convex potential and one by stress diffusion. We show short-time existence of strong solutions as well as their uniqueness in a class of Leray-Hopf-type weak solutions satisfying the tensorial component in the sense of an evolutionary variational inequality. The global-in-time existence of such generalized solutions has been established in a previous work. We further study the limit when stress diffusion vanishes. In this case, the above notion of generalized solutions is no longer suitable, and we introduce the concept of energy-variational solutions, which is based on an inequality for the relative energy. We derive general properties of energy-variational solutions and show their existence by passing to the nondiffusive limit in the relative energy inequality satisfied by generalized solutions for nonzero stress diffusion.

Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits

We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. These solutions are weak in the probabilistic sense i.e., the probability space and the driving Wiener process are an integral part of the solution. We derive the relative energy inequality for the stochastic compressible Euler equations and, as a corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use of the relative energy inequality, we investigate the low Mach limit (incompressible limit) of underlying system of equations. As a main novelty with respect to the related literature, our results apply to general nonlinear multiplicative stochastic perturbations of Nemytskij type.

Weak–strong uniqueness for heat conducting non-Newtonian incompressible fluids

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 balance of energy and has the advantage that it admits a weak–strong uniqueness principle, justifying the proposed formulation. We provide a proof of existence of solutions based on finite element approximations, thus obtaining the first convergence result of a numerical scheme for the full evolutionary system including temperature dependent coefficients and viscous dissipation terms. Then we proceed to prove the weak–strong uniqueness property of the system by means of a relative energy inequality.

On Weak-Strong Uniqueness for Stochastic Equations of Incompressible Fluid Flow

We introduce a concept of dissipative measure-valued martingale solution to the stochastic Euler equations describing the motion of an inviscid incompressible fluid. These solutions are characterized by a parametrized Young measure and a concentration defect measure in the total energy balance. Moreover, they are weak in the probabilistic sense i.e., the underlying probability space and the driving Wiener process are intrinsic parts of the solution. We first exhibit the relative energy inequality for the incompressible Euler equations driven by a multiplicative noise and then demonstrate the pathwise weak-strong uniqueness principle. Finally, we also provide a sufficient condition, á la Prodi (Ann Mat Pura Appl 48:173–182, 1959) and Serrin (in: Nonlinear problems, University of Wisconsin Press, Madison, Wisconsin, pp 69–98, 1963), for the uniqueness of weak martingale solutions to the stochastic Navier–Stokes system in the class of finite energy weak martingale solutions.

On weak (measure-valued)-strong uniqueness for compressible MHD system with non-monotone pressure law

<p style='text-indent:20px;'>In this paper, we define a renormalized dissipative measure-valued (rDMV) solution of the compressible magnetohydrodynamics (MHD) equations with non-monotone pressure law. We prove the existence of the rDMV solutions and establish a suitable relative energy inequality. And we obtain the weak (measure-valued)-strong uniqueness property of this rDMV solution with the help of the relative energy inequality.</p>

Existence and stability of dissipative turbulent solutions to a simple bi-fluid model of compressible fluids

Following Abbatiello et al. (DCCDS-A 41(1):1–28, 2020), we introduce dissipative turbulent solutions to a simple model of a mixture of two non interacting compressible fluids filling a bounded domain with general non zero inflow/outflow boundary conditions. We prove existence of such solutions for all adiabatic coefficients \(\gamma >1\), their compatibility with classical solutions, the relative energy inequality, and the weak–strong uniqueness principle in this class. The class of dissipative turbulent solutions is so far the largest class of generalized solutions which still enjoys the weak–strong uniqueness property.

Analysis of a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model

In this paper, existence of generalized solutions to a thermodynamically consistent Navier–Stokes–Cahn–Hilliard model introduced in Eleuteri et al. (2015) is proven in any space dimension. The generalized solvability concepts are measure-valued and dissipative solutions. The measure-valued formulation incorporates an entropy inequality and an energy inequality instead of an energy balance in a nowadays standard way, the Gradient flow of the internal variable is fulfilled in a weak and the momentum balance in a measure-valued sense. In the dissipative formulation, the distributional relations of the momentum balance and the energy as well as entropy inequality are replaced by a relative energy inequality. Additionally, we prove the weak–strong uniqueness of the proposed solution concepts and that all generalized solutions with additional regularity are indeed strong solutions.

Navier–Stokes–Fourier System with General Boundary Conditions

We consider the Navier--Stokes--Fourier system in a bounded domain $\Omega \subset R^d$, $d=2,3$, with physically realistic in/out flow boundary conditions. We develop a new concept of weak solutions satisfying a general form of relative energy inequality. The weak solutions exist globally in time for any finite energy initial data and comply with the weak--strong uniqueness principle.

Dissipative Solutions to Compressible Navier–Stokes Equations with General Inflow–Outflow Data: Existence, Stability and Weak Strong Uniqueness

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., in particular, when the normal component of the velocity on the boundary of the flow domain is equal to zero). Most of physical applications (as flows in wind tunnels, pipes, reactors of jet engines) requires to consider non-zero inflow–outflow boundary condtions. We prove existence of dissipative weak solutions to the compressible Navier–Stokes equations in barotropic regime (adiabatic coefficient \(\gamma >3/2\), in three dimensions, \(\gamma >1\) in two dimensions) with large velocity prescribed at the boundary and large density prescribed at the inflow boundary of a bounded piecewise regular Lipschitz domain, without any restriction neither on the shape of the inflow/outflow boundaries nor on the shape of the domain. It is well known that the relative energy inequality has many applications, e.g., to investigation of incompressible or inviscid limits, to the dimension reduction of flows, to the error estimates of numerical schemes. In this paper we deal with one of its basic applications, namely weak–strong uniqueness principle.

Dissipative martingale solutions of the stochastically forced Navier–Stokes–Poisson system on domains without boundary

We construct solutions to the randomly-forced Navier–Stokes–Poisson system in periodic three-dimensional domains or in the whole three-dimensional Euclidean space. These solutions are weak in the sense of PDEs and also weak in the sense of probability. As such, they satisfy the system in the sense of distributions and the underlying probability space and the stochastic driving force are also unknowns of the problem. Additionally, these solutions dissipate energy, satisfies a relative energy inequality in the sense of Dafermos (1979) and satisfy a renormalized form of the continuity equation in the sense of DiPerna and Lions (1989).

On Weak (Measure-Valued)\u2013Strong Uniqueness for Compressible Navier\u2013Stokes System with Non-monotone Pressure Law

In this paper our goal is to define a renormalized dissipative measure-valued (rDMV) solution of compressible Navier–Stokes system for fluids with non-monotone pressure–density relation. We prove existence of rDMV solutions and establish a suitable relative energy inequality. Moreover we obtain the weak (measure-valued)–strong uniqueness property of this rDMV solution with the help of relative energy inequality.

Weak-strong uniqueness for the compressible fluid-rigid body interaction

In this work we study the coupled system of partial and ordinary differential equations describing the interaction between a compressible isentropic viscous fluid and a rigid body moving freely inside the fluid. In particular the position and velocity of the rigid body in the fluid are unknown and the motion of the rigid body is driven by the normal stress forces of the fluid acting on the boundary of the body. We prove that the strong solution, which is known to exist under certain smallness assumptions, is unique in the class of weak solutions to the problem. The proof relies on a correct definition of the relative energy, to use this tool we then have to introduce a change of coordinates to transform the strong solution to the domain of the weak solution in order to use it as a test function in the relative energy inequality. Estimating all arising terms we prove that the weak solution has to coincide with the transformed strong solution and finally that the transformation has to be in fact an identity.

Weak-strong uniqueness for a bi-fluid model for a mixture of non-interacting compressible fluids

We investigate a version of one velocity Baer-Nunziato type system with dissipation describing the motion of a mixture of two compressible fluids. We define for this system weak solutions on one hand and dissipative weak solutions on the other hand, and recall the theorem about their existence on a large time interval. We investigate strong solutions and show their existence on a short time interval. Finally, we prove that any weak solution satisfies a relative energy inequality and prove for this system the weak-strong uniqueness principle. This is the main result of the paper.

On weak–strong uniqueness for compressible Navier–Stokes system with general pressure laws

The goal of the present paper is to study the weak–strong uniqueness problem for the compressible Navier–Stokes system with a general barotropic pressure law. Our results include the case of a hard sphere pressure of Van der Waals type with a non-monotone perturbation and a Lipschitz perturbation of a monotone pressure. Although the main tool is the relative energy inequality, the results are conditioned by the presence of viscosity and do not seem extendable to the Euler system.

Stability of strong solutions for a model of incompressible two–phase flow under thermal fluctuations

We consider a model of a two–phase flow based on the phase field approach, where the fluid bulk velocity obeys the standard Navier–Stokes system while the concentration difference of the two fluids plays a role of order parameter governed by the Allen–Cahn equations. Possible thermal fluctuations are incorporated through a random forcing term in the Allen–Cahn equation. We show that suitable dissipative martingale solutions satisfy a stochastic version of the relative energy inequality. This fact is used for showing the weak–strong uniqueness principle both pathwise and in law.