Dissipative Weak Solutions Research Articles

Existence of energy-variational solutions to hyperbolic conservation laws

We introduce the concept of energy-variational solutions for hyperbolic conservation laws. Intrinsically, these energy-variational solutions fulfill the weak-strong uniqueness principle and the semi-flow property, and the set of solutions is convex and weakly-star closed. The existence of energy-variational solutions is proven via a suitable time-discretization scheme under certain assumptions. This general result yields existence of energy-variational solutions to the magnetohydrodynamical equations for ideal incompressible fluids and to the Euler equations in both the incompressible and the compressible case. Moreover, we show that energy-variational solutions to the Euler equations coincide with dissipative weak solutions.

The role of the pressure in the regularity theory for the Navier-Stokes equations

We first show the equivalence of two classes of generalized suitable weak solutions to the 3D incompressible Navier-Stokes equations allowing distributional pressure, the class of dissipative weak solutions and local suitable weak solutions. Then, an $\varepsilon$-regularity criterion for dissipative weak solutions follows from that for local suitable weak solutions. We relax the $\varepsilon$-regularity criterion with a new approach using a local version of Leray projection operator. As an application of the approach, we obtain the short-time regularity result on a bounded domain for dissipative solutions.

Nonlinear open mapping principles, with applications to the Jacobian equation and other scale-invariant PDEs

For a nonlinear operator T satisfying certain structural assumptions, our main theorem states that the following claims are equivalent: i) T is surjective, ii) T is open at zero, and iii) T has a bounded right inverse. The theorem applies to numerous scale-invariant PDEs in regularity regimes where the equations are stable under weak⁎ convergence. Two particular examples we explore are the Jacobian equation and the equations of incompressible fluid flow.For the Jacobian, it is a long standing open problem to decide whether it is onto between the critical Sobolev space and the Hardy space. Towards a negative answer, we show that, if the Jacobian is onto, then it suffices to rule out the existence of surprisingly well-behaved solutions.For the incompressible Euler equations, we show that, for any p<∞, the set of initial data for which there are dissipative weak solutions in LtpLx2 is meagre in the space of solenoidal L2 fields. Similar results hold for other equations of incompressible fluid dynamics.

On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions

In this work, we prove the convergence of residual distribution (RD) schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the RD schemes are fulfilling the underlying structure preserving methods properties such as positivity of density and internal energy. Consequently, the RD schemes lead to a consistent and stable approximation of the Euler equations. Our result can be seen as a generalization of the Lax–Richtmyer equivalence theorem to nonlinear problems that consistency plus stability is equivalent to convergence.

Convergence of discontinuous Galerkin schemes for the Euler equations via dissipative weak solutions

In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving properties, such as positivity preservation and entropy inequality hold. We demonstrate how to ensure them and prove the convergence of our multidimensional high-order DG scheme via dissipative weak solutions. In numerical simulations, we verify our theoretical results.

On convergence of numerical solutions for the compressible MHD system with weakly divergence-free magnetic field

Abstract We study a general convergence theory for the analysis of numerical solutions to a magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension $d$$(=2,3)$. First, we introduce the concept of dissipative weak (DW) solutions and prove the weak–strong uniqueness property for DW solutions, meaning a DW solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of consistent approximations towards the DW solution, as well as the classical solution. Interpreting the consistent approximation as the energy stability and consistency of numerical solutions, we have built a nonlinear variant of the celebrated Lax equivalence theorem. Finally, as an application of this theory, we show the convergence analysis of two numerical methods.

A Numerical Approach for the Existence of Dissipative Weak Solutions to a Compressible Two-fluid Model

A Numerical Approach for the Existence of Dissipative Weak Solutions to a Compressible Two-fluid Model

Strong solutions of a stochastic differential equation with irregular random drift

We present a well-posedness result for strong solutions of one-dimensional stochastic differential equations (SDEs) of the form dX=u(ω,t,X)dt+12σ(ω,t,X)∂xσ(ω,t,X)dt+σ(ω,t,X)dW(t),where the drift coefficient u is random and irregular, with a weak derivative satisfying ∂xu=q for some q∈LωpLt∞(Lx2∩Lx1), p∈[1,∞). The random and regular noise coefficient σ may vanish. The main contribution is a pathwise uniqueness result under the assumptions that E‖q(t)−q(0)‖L2(R)2→0 as t↓0, and u satisfies the one-sided gradient bound q(ω,t,x)≤K(ω,t), where the process K(ω,t)>0 exhibits an exponential moment bound of the form Eexp(p∫tTK(s)ds)≲t−2p for small times t, for some p≥1. This study is motivated by ongoing work on the well-posedness of the stochastic Hunter–Saxton equation, a stochastic perturbation of a nonlinear transport equation that arises in the modelling of the director field of a nematic liquid crystal. In this context, the one-sided bound acts as a selection principle for dissipative weak solutions of the stochastic partial differential equation.

Self-regularization in turbulence from the Kolmogorov 4/5-law and alignment.

A defining feature of three-dimensional hydrodynamic turbulence is that the rate of energy dissipation is bounded away from zero as viscosity is decreased (Reynolds number increased). This phenomenon-anomalous dissipation-is sometimes called the 'zeroth law of turbulence' as it underpins many celebrated theoretical predictions. Another robust feature observed in turbulence is that velocity structure functions [Formula: see text] exhibit persistent power-law scaling in the inertial range, namely [Formula: see text] for exponents [Formula: see text] over an ever increasing (with Reynolds) range of scales. This behaviour indicates that the velocity field retains some fractional differentiability uniformly in the Reynolds number. The Kolmogorov 1941 theory of turbulence predicts that [Formula: see text] for all [Formula: see text] and Onsager's 1949 theory establishes the requirement that [Formula: see text] for [Formula: see text] for consistency with the zeroth law. Empirically, [Formula: see text] and [Formula: see text], suggesting that turbulent Navier-Stokes solutions approximate dissipative weak solutions of the Euler equations possessing (nearly) the minimal degree of singularity required to sustain anomalous dissipation. In this note, we adopt an experimentally supported hypothesis on the anti-alignment of velocity increments with their separation vectors and demonstrate that the inertial dissipation provides a regularization mechanism via the Kolmogorov 4/5-law. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

Weak–strong uniqueness for a class of generalized dissipative weak solutions for non-homogeneous, non-Newtonian and incompressible fluids

We consider a system of PDEs describing a non-homogeneous, non-Newtonian and incompressible fluid in a space-periodic domain. We define the generalized dissipative weak solutions as well as give a proof of its existence, when the boundary data is provided. The main result of this work is the weak–strong uniqueness of the defined solutions.

On the uniqueness of solutions to hyperbolic systems of conservation laws

For general hyperbolic systems of conservation laws we show that dissipative weak solutions belonging to an appropriate Besov space B q α , ∞ and satisfying a one-sided bound condition are unique within the class of dissipative solutions. The exponent α > 1 / 2 is universal independently of the nature of the nonlinearity and the Besov regularity need only be imposed in space when the system is expressed in appropriate variables. The proof utilises a commutator estimate which allows for an extension of the relative entropy method to the required regularity setting. The systems of elasticity, shallow water magnetohydrodynamics, and isentropic Euler are investigated, recovering recent results for the latter. Moreover, the article explores a triangular system motivated by studies in chromatography and constructs an explicit solution which fails to be Lipschitz, yet satisfies the conditions of the presented uniqueness result.

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.

Semiflow selection for the compressible Navier\u2013Stokes system

Although the existence of dissipative weak solutions for the compressible Navier–Stokes system has already been established for any finite energy initial data, uniqueness is still an open problem. The idea is then to select a solution satisfying the semigroup property, an important feature of systems with uniqueness. More precisely, we are going to prove the existence of a semiflow selection in terms of the three state variables: the density, the momentum, and the energy. Finally, we will show that it is possible to introduce a new selection defined only in terms of the initial density and momentum; however, the price to pay is that the semigroup property will hold almost everywhere in time.

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.

Nonuniqueness of weak solutions to the Navier-Stokes equation

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Holder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

Weak solutions for Euler systems with non-local interactions

We consider several modifications of the Euler system of fluid dynamics, including its pressureless variant driven by non-local interaction repulsive–attractive and alignment forces in the space dimension N = 2 , 3 . These models arise in the study of self-organization in collective behavior modeling of animals and crowds. We adapt the method of convex integration to show the existence of infinitely many global-in-time weak solutions for any bounded initial data. Then we consider the class of dissipative solutions satisfying, in addition, the associated global energy balance (inequality). We identify a large set of initial data for which the problem admits infinitely many dissipative weak solutions. Finally, we establish a weak–strong uniqueness principle for the pressure-driven Euler system with non-local interaction terms as well as for the pressureless system with Newtonian interaction.

Global existence of weak solutions for a nonlocal model for two-phase flows of incompressible fluids with unmatched densities

We consider a diffuse interface model for an incompressible isothermal mixture of two viscous Newtonian fluids with different densities in a bounded domain in two or three space dimensions. The model is the nonlocal version of the one recently derived by Abels, Garcke and Grün and consists in a Navier–Stokes type system coupled with a convective nonlocal Cahn–Hilliard equation. The density of the mixture depends on an order parameter. For this nonlocal system we prove existence of global dissipative weak solutions for the case of singular double-well potentials and non-degenerate mobilities. To this goal we devise an approach which is completely independent of the one employed by Abels, Depner and Garcke to establish existence of weak solutions for the local Abels et al. model.

Dissipative weak solutions to compressible Navier–Stokes–Fokker–Planck systems with variable viscosity coefficients

Motivated by a recent paper by Barrett and Süli (2016) [6], we consider the compressible Navier–Stokes system coupled with a Fokker–Planck type equation describing the motion of polymer molecules in a viscous compressible fluid occupying a bounded spatial domain, with polymer-number-density-dependent viscosity coefficients. The model arises in the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The motion of the solvent is governed by the unsteady, compressible, barotropic Navier–Stokes system, where the viscosity coefficients in the Newtonian stress tensor depend on the polymer number density. Our goal is to show that the existence theory developed in the case of constant viscosity coefficients can be extended to the case of polymer-number-density-dependent viscosities, provided that certain technical restrictions are imposed, relating the behavior of the viscosity coefficients and the pressure for large values of the solvent density. As a first step in this direction, we prove here the weak sequential stability of the family of dissipative (finite-energy) weak solutions to the system.

Towards a parameter-free method for high Reynolds number turbulent flow simulation based on adaptive finite element approximation

This article is a review of our work towards a parameter-free method for simulation of turbulent flow at high Reynolds numbers. In a series of papers we have developed a model for turbulent flow in the form of weak solutions of the Navier–Stokes equations, approximated by an adaptive finite element method , where: (i) viscous dissipation is assumed to be dominated by turbulent dissipation proportional to the residual of the equations, and (ii) skin friction at solid walls is assumed to be negligible compared to inertial effects. The result is a computational model without empirical data, where the only model parameter is the local size of the finite element mesh . Under adaptive refinement of the mesh based on a posteriori error estimation, output quantities of interest in the form of functionals of the finite element solution converge to become independent of the mesh resolution, and thus the resulting method has no adjustable parameters. No ad hoc design of the mesh is needed, instead the mesh is optimized based on solution features, in particular no boundary layer mesh is needed. We connect the computational method to the mathematical concept of a dissipative weak solution of the Euler equations , as a model of high Reynolds number turbulent flow, and we highlight a number of benchmark problems for which the method is validated. The purpose of the article is to present the computational framework in a concise form , to report on recent progress, and to discuss open problems that are subject to ongoing research. • We present a parameter-free method for high Reynolds number turbulent flow. • The method is based on adaptive mesh optimization using adjoint techniques. • We connect the numerical approximation to dissipative weak solutions. • Turbulent boundary layers are left unresolved, no boundary layer mesh is needed. • We highlight benchmark problems for which the method is validated.

Well/Ill Posedness for the Euler-Korteweg-Poisson System and Related Problems

We consider a general Euler-Korteweg-Poisson system in R 3, supplemented with the space periodic boundary conditions, where the quantum hydrodynamics equations and the classical fluid dynamics equations with capillarity are recovered as particular examples. We show that the system admits infinitely many global-in-time weak solutions for any sufficiently smooth initial data including the case of a vanishing initial density - the vacuum zones. Moreover, there is a vast family of initial data, for which the Cauchy problem possesses infinitely many dissipative weak solutions, i.e. the weak solutions satisfying the energy inequality. Finally, we establish the weak-strong uniqueness property in a class of solutions without vacuum. In this paper we show that, even in presence of a dispersive tensor, we have the same phenomena found by De Lellis and Székelyhidi.