Entropy Weak Solutions Research Articles

Weak entropy solution for a Keller-Segel type fluid model

In this paper, we consider a Keller-Segel type fluid model, which is a kind of Euler-Poisson system with a self-gravitational force. We show that similar to the parabolic case, there is a critical mass $8\pi$ such that if the initial total mass $M$ is supercritical, i.e., $M> 8\pi$, then any weak entropy solution with the same mass $M$ must blow up in finite time. The a priori estimates of weak entropy solutions for critical mass $M=8\pi$ and subcritical mass $M<8\pi$ are also obtained.

Global existence of entropy-weak solutions to the compressible Navier–Stokes equations with non-linear density dependent viscosities

In this paper, we considerably extend the results on global existence of entropy-weak solutions to the compressible Navier–Stokes system with density dependent viscosities obtained, independently (using different strategies) by Vasseur–Yu [Invent. Math. 206 (2016) and arXiv:1501.06803 (2015)] and by Li–Xin [arXiv:1504.06826 (2015)]. More precisely, we are able to consider a physical symmetric viscous stress tensor \sigma = 2 \mu(\rho) \,{\mathbb D}(u) +(\lambda(\rho) \operatorname{div} u - P(\rho) \operatorname {Id} where {\mathbb D}(u) = [\nabla u + \nabla^T u]/2 with shear and bulk viscosities (respectively \mu(\rho) and \lambda(\rho) ) satisfying the BD relation \lambda(\rho)=2(\mu'(\rho)\rho - \mu(\rho)) and a pressure law P(\rho)=a\rho^\gamma (with a&gt;0 a given constant) for any adiabatic constant \gamma&gt;1 . The non-linear shear viscosity \mu(\rho) satisfies some lower and upper bounds for low and high densities (our result includes the case \mu(\rho)= \mu\rho^\alpha with 2/3 &lt; \alpha &lt; 4 and \mu&gt;0 constant). This provides an answer to a longstanding question on compressible Navier–Stokes equations with density dependent viscosities, mentioned for instance by F. Rousset [Bourbaki 69ème année, 2016–2017, exp. 1135].

Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and ℝ, and v is valued in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D and 3D cases where the solution does not belong to H1 (Ω), show that this method can provide accurate results. We then construct a numerical scheme which presents a convergence property to the entropy weak solution of the problem in the case where the right-hand side belongs to L1. This is achieved owing to a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation, and adding an upstream weighting evaluation and a nonlinear p-Laplace vanishing stabilisation term.

Large Eddy Simulations by approximate weak entropy solutions

We propose to use weak (averaged) entropy solutions in lieu of LES models. This approach unites the theory for shock capturing schemes and turbulence modelling. To achieve this, we identify a number of conditions (albeit not sufficient) that a scheme should satisfy. Namely, a scheme should be: conservative, entropy dissipative, kinetic-energy preserving/diffusive, positivity preserving and have linearly stable (non anti-diffusive) continuity equation. We propose a finite-volume scheme with these properties and investigate its properties, and the properties of some related schemes, for the standard entropy-wave/shock interaction, a Kelvin-Helmoholtz instability, and a turbulent Rayleigh-Taylor problem.These preliminary investigations suggest that the scheme is very robust, is competitive for turbulent problems and is far less prone to trip false turbulence. However, and as with any general purpose scheme, wildly under-resolved simulations can not be expected to be accurate. The advantage with the current scheme is that local averages converge which provides a possibility to estimate the accuracy of functionals of interest.

Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ordinary differential equation (ODE) for the different phases of steel, and Maxwell’s equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e. that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g. it allows to include free energy functions with low regularity properties corresponding to phase transitions.

Quasi-static limit for a hyperbolic conservation law

We study the quasi-static limit for the $L^\infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with the quasi-static equation, whose entropy solution is determined by the stationary profile corresponding to the boundary data at a given time.

Singular diffusion with Neumann boundary conditions

In this paper we develop an existence theory for the nonlinear initial-boundary value problem with singular diffusion , u| t=0 = u 0 with Neumann boundary conditions k(x)∇G(u) ⋅ ν = 0. Here , a bounded open set with C 3 boundary, and with ν as the unit outer normal. The function G is Lipschitz continuous and nondecreasing, while k(x) is a diagonal matrix. We show that any two weak entropy solutions u and v satisfy , for almost every t ⩾ 0, and a constant C = C(k, G, B). If we restrict to the case when the entries k i of k depend only on the corresponding component, k i = k i (x i ), and ∂B is C 2, we show that there exists an entropy solution, thus establishing in this case that the problem is well-posed in the sense of Hadamard.

SBV regularity for Burgers-Poisson equation

The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in L1(R) is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.

Global Bounded Weak Entropy Solutions to the Euler--Vlasov Equations in Fluid-Particle System

In this paper, we consider a fluid-particle system which describes the evolution of a two-phase flow. The system consists of the compressible Euler equations for the fluid (fluid phase) coupled with the Vlasov equation for the particles (disperse phase) through the drag force. We obtain a global bounded weak entropy solution for such one-dimensional Euler--Vlasov equations with arbitrarily large initial data for the whole range of physical adiabatic exponents $\gamma>1$. To achieve this, we apply the vanishing viscosity method and compensated compactness theory. We construct globally defined approximate solutions by adding our novel viscosity terms to the Euler equations, which together with our key observation on relative velocity plays a fundamental role in the hardest part of our proof: the uniform $L^\infty$ estimate. After deriving an entropy dissipation estimate, we prove the convergence of approximate solutions by the compensated compactness argument.

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

Existence and uniqueness results for a class of nonlocal conservation laws by means of a Lax–Hopf-type solution formula

We study the initial value problem and the initial boundary value problem for nonlocal conservation laws. The nonlocal term is realized via a spatial integration of the solution between specified boundaries and affects the flux function of a given “local” conservation law in a multiplicative way. For a strictly convex flux function and strictly positive nonlocal impact we prove existence and uniqueness of weak entropy solutions relying on a fixed-point argument for the nonlocal term and an explicit Lax–Hopf-type solution formula for the corresponding Hamilton–Jacobi (HJ) equation. Using the developed theory for HJ equations, we obtain a semi-explicit Lax–Hopf-type formula for the solution of the corresponding nonlocal HJ equation and a semi-explicit Lax–Oleinik-type formula for the nonlocal conservation law.

Convergence of time-splitting approximations for degenerate convection-diffusion equations with a random source

AbstractIn this paper we propose a time-splitting method for degenerate convection-diffusion equations perturbed stochastically by white noise. This work generalizes previous results on splitting operator techniques for stochastic hyperbolic conservation laws for the degenerate parabolic case. The convergence inLlocp$\begin{array}{} \displaystyle L^p_{loc} \end{array}$of the time-splitting operator scheme to the unique weak entropy solution is proven. Moreover, we analyze the performance of the splitting approximation by computing its convergence rate and showing numerical simulations for some benchmark examples, including a fluid flow application in porous media.

The entropy weak solution to a generalized Fornberg\u2013Whitham equation

We investigate a nonlinear generalized Fornberg–Whitham equation. The key element is that we derive an L^{2}(mathbb{R}) conservation law for solutions of the equation. We establish several estimates by utilizing the L^{2}(mathbb{R}) conservation law. These estimates lead to the proof of the existence and uniqueness of entropy weak solution of the equation in the space L^{1}(mathbb{R})cap L^{infty}(mathbb{R}).

Existence and uniqueness of the entropy solution of a stochastic conservation law with a Q‐Brownian motion

In this paper, we prove the existence and uniqueness of the entropy solution for a first‐order stochastic conservation law with a multiplicative source term involving a ‐Brownian motion. After having defined a measure‐valued weak entropy solution of the stochastic conservation law, we present the Kato inequality, and as a corollary, we deduce the uniqueness of the measure‐valued weak entropy solution, which coincides with the unique weak entropy solution of the problem. The Kato inequality is proved by a doubling of variables method; to that purpose, we prove the existence and the uniqueness of the strong solution of an associated stochastic nonlinear parabolic problem by means of an implicit time discretization scheme; we also prove its convergence to a measure‐valued entropy solution of the stochastic conservation law, which proves the existence of the measure‐valued entropy solution.

On a limit of perturbed conservation laws with saturating diffusion and non-positive dispersion

We consider a conservation law with convex flux, perturbed by a saturating diffusion and non-positive dispersion of the form $$u_t + f(u)_x = \varepsilon \Big ( {u_{x}\over \sqrt{1 + u_x^2}} \Big )_x - \delta (|u_{xx}|^n)_x$$. We prove the convergence of the solutions $$\{u^{\varepsilon , \delta }\}$$ to the entropy weak solution of the hyperbolic conservation law, $$u_t + f(u)_x = 0,$$ for all real number $$1\le n \le 2$$ provided $$\delta = o(\varepsilon ^{n(n+1)\over 2}; \varepsilon ^{n+1\over n })$$.

A PDE model for the spatial dynamics of a voles population structured in age

We prove existence and stability of entropy weak solutions for a macroscopic PDE model for the spatial dynamics of a population of voles structured in age. The model consists of a scalar PDE depending on time, t, age, a, and space x=(x1,x2), supplemented with a non-local boundary condition at a=0. The flux is linear with constant coefficient in the age direction but contains a non-local term in the space directions. Also, the equation contains a term of second order in the space variables only. Existence of solutions is established by compensated compactness, see Panov (2009), and we prove stability by a doubling of variables type argument.

On oscillatory solutions to the complete Euler system

The Euler system in fluid dynamics is a model of a compressible inviscid fluid incorporating the three basic physical principles: Conservation of mass, momentum, and energy. We show that the Cauchy problem is basically ill-posed for the L∞-initial data in the class of weak entropy solutions. As a consequence, there are infinitely many measure-valued solutions for a vast set of initial data. Finally, using the concept of relative energy, we discuss a singular limit problem for the measure-valued solutions, where the Mach and Froude number are proportional to a small parameter.

Convergence of a family of perturbed conservation laws with diffusion and non-positive dispersion

We consider a family of conservation laws with convex flux perturbed by vanishing diffusion and non-positive dispersion of the form ut+f(u)x=εuxx−δ(|uxx|n)x. Convergence of the solutions {uε,δ} to the entropy weak solution of the hyperbolic limit equation ut+f(u)x=0, for all real numbers 1≤n≤2 is proved if δ=o(ε3n−12;ε5n−12(2n−1)).

Convergence of the finite volume method on a Schwarzschild background

We introduce a class of nonlinear hyperbolic conservation laws on a Schwarzschild black hole background and derive several properties satisfied by (possibly weak) solutions. Next, we formulate a numerical approximation scheme which is based on the finite volume methodology and takes the curved geometry into account. An interesting feature of our model is that no boundary conditions is required at the black hole horizon boundary. We establish that this scheme converges to an entropy weak solution to the initial value problem and, in turn, our analysis also provides us with a theory of existence and stability for a new class of conservation laws.

A cell-centred pressure-correction scheme for the compressible Euler equations

Abstract We propose a robust pressure-correction scheme for the numerical solution of the compressible Euler equations discretized by a collocated finite volume method. The scheme is based on an internal energy formulation, which ensures that the internal energy is positive. More generally, the scheme enjoys fundamental stability properties: without restriction on the time step, both the density and the internal energy are positive, the integral of the total energy over the computational domain is preserved thanks to an estimate on the discrete kinetic energy and a discrete entropy inequality is satisfied. These stability properties ensure the existence of a solution to the scheme. The internal energy balance features a corrective source term, which is needed for the scheme to compute the correct shock solutions; we are indeed able to prove a Lax-consistency-type convergence result, in the sense that, under some compactness assumptions, the limit of a converging sequence of approximate solutions obtained with space and time discretization steps tending to zero is an entropy weak solution of the Euler equations. Moreover, constant pressure and velocity are preserved through contact discontinuities. The obtained theoretical results and the scheme accuracy are verified by numerical experiments; a numerical stabilization is introduced in order to reduce the oscillations that appear for some tests. The qualitative behaviour of the scheme is assessed on one-dimensional and two-dimensional Riemann problems and compared with other schemes.