Entropy Weak Solutions Research Articles

Discrete shock profiles for scalar conservation laws with discontinuous fluxes

The paper considers a scalar conservation law with a discontinuous flux F of the form F(x,u)=H(x)g(u)+(1−H(x))h(u) where H(x) is the Heaviside function. Herein, the fluxes g and h are supposed to have one minimum and no maximum and at most one crossing in the interior of the domain of definition. The aim is to verify a weak solution of such a problem in the following way: We are looking for discrete shock profiles for its continuously differentiable perturbation with a parameter ε and Godunov's scheme for conservation laws with spatially varying flux functions. The obtained discrete shock profile satisfies a discrete entropy condition of Kruzkhov type and after letting ε→0, approaches an entropy weak solution of the original equation.

Relative entropy for compressible Navier–Stokes equations with density-dependent viscosities and applications

Recently A. Vasseur and C. Yu have proved (see A. Vasseur, C. Yu, Existence of global weak solutions for 3D degenerate compressible Navier–Stokes equations, arXiv:1501.06803, 2015) the existence of global entropy-weak solutions to the compressible Navier–Stokes equations with viscosities ν(ϱ)=μϱ and λ(ϱ)=0 and a pressure law under the form p(ϱ)=aϱγ with a>0 and γ>1 constants. In this note, we propose a non-trivial relative entropy for such system in a periodic box and give some applications. This extends, in some sense, results with constant viscosities recently initiated by E. Feireisl, B.J. Jin and A. Novotny in [J. Math. Fluid Mech. (2012)]. We present some mathematical results related to the weak–strong uniqueness, the convergence to a dissipative solution to compressible or incompressible Euler equations. As a by-product, this mathematically justifies the convergence of solutions to a viscous shallow-water system to solutions to the inviscid shallow-water system.

The Compressible to Incompressible Limit of One Dimensional Euler Equations: The Non Smooth Case

We prove a rigorous convergence result for the compressible to incompressible limit of weak entropy solutions to the isothermal one dimensional Euler equations.

Mixtures: Sequential Stability of Variational Entropy Solutions

The purpose of this work is to analyze the mathematical model governing motion of n-component, heat conducting reactive mixture of compressible gases. We prove sequential stability of weak variational entropy solutions when the state equation essentially depends on the species concentration and the species diffusion fluxes depend on gradients of partial pressures. Of crucial importance for our analysis is the fact that viscosity coefficients vanish on vacuum and the source terms enjoy the admissibility condition dictated by the second law of thermodynamics.

Relative entropy and compressible potential flow

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density ▪ and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ,v if and only if |v|<c. For motivation some simple variations on the relative entropy theme of Dafermos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible perturbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.

Well-posedness of a conservation law with non-local flux arising in traffic flow modeling

We prove the well-posedness of entropy weak solutions of a scalar conservation law with non-local flux arising in traffic flow modeling. The result is obtained providing accurate $$\mathbf {L^\infty }$$L?, BV and $$\mathbf {L^1}$$L1 estimates for the sequence of approximate solutions constructed by an adapted Lax-Friedrichs scheme.

On some implicit and semi-implicit staggered schemes for the shallow water and Euler equations

In this paper, we propose implicit and semi-implicit in time finite volume schemes for the barotropic Euler equations (hence, as a particular case, for the shallow water equations) and for the full Euler equations, based on staggered discretizations. For structured meshes, we use the MAC finite volume scheme, and, for general mixed quadrangular/hexahedral and simplicial meshes, we use the discrete unknowns of the Rannacher−Turek or Crouzeix−Raviart finite elements. We first show that a solution to each of these schemes satisfies a discrete kinetic energy equation. In the barotropic case, a solution also satisfies a discrete elastic potential balance; integrating these equations over the domain readily yields discrete counterparts of the stability estimates which are known for the continuous problem. In the case of the full Euler equations, the scheme relies on the discretization of the internal energy balance equation, which offers two main advantages: first, we avoid the space discretization of the total energy, which involves cell-centered and face-centered variables; second, we obtain an algorithm which boils down to a usual pressure correction scheme in the incompressible limit. Consistency (in a weak sense) with the original total energy conservative equation is obtained thanks to corrective terms in the internal energy balance, designed to compensate numerical dissipation terms appearing in the discrete kinetic energy inequality. It is then shown in the 1D case, that, supposing the convergence of a sequence of solutions, the limit is an entropy weak solution of the continuous problem in the barotropic case, and a weak solution in the full Euler case. Finally, we present numerical results which confirm this theory.

Oleinik type estimates for the Ostrovsky–Hunter equation

The Ostrovsky–Hunter equation provides a model of small-amplitude long waves in a rotating fluid of finite depth. This is a nonlinear evolution equation. In this study, we consider the well-posedness of the Cauchy problem associated with this equation within a class of bounded discontinuous solutions. We show that we can replace the Kruzkov-type entropy inequalities with an Oleinik-type estimate and we prove the uniqueness via a nonlocal adjoint problem. This implies that a shock wave in an entropy weak solution to the Ostrovsky–Hunter equation is admissible only if it jumps down in value (similar to the inviscid Burgers' equation).

On compactness estimates for hyperbolic systems of conservation laws

We study the compactness in Lloc1 of the semigroup mapping (St)t>0 defining entropy weak solutions of general hyperbolic systems of conservation laws in one space dimension. We establish a lower estimate for the Kolmogorov ε-entropy of the image through the mapping St of bounded sets in L1∩L∞, which is of the same order 1/ε as the ones established by the authors for scalar conservation laws. We also provide an upper estimate of order 1/ε for the Kolmogorov ε-entropy of such sets in the case of Temple systems with genuinely nonlinear characteristic families, that extends the same type of estimate derived by De Lellis and Golse for scalar conservation laws with convex flux. As suggested by Lax, these quantitative compactness estimates could provide a measure of the order of “resolution” of the numerical methods implemented for these equations.

Erratum to: On entropy weak solutions of Hughes’ model for pedestrian motion

In the original publication of the article, the article note at the end of the title page was incorrect. The correct article note is given below: This research was supported by the ERC Starting Grant 2010 under the project “TRAffic Management by Macroscopic Models” and the Polonium 2011 (French-Polish cooperation program) under the project “CROwd Motion Modeling and Management” No. 8424/2011. Projekt zosta l sfinansowany ze środkow Narodowego Centrum Nauki przyznanych na podstawie decyzji nr: DEC-2011/01/B/ST1/03965.

Multi-dimensional scalar balance laws with discontinuous flux

We consider scalar balance laws with a dissipative source term. The flux function may be discontinuous with respect to both the space variable x and the unknown quantity u. We formulate the definition of entropy weak solutions and provide existence and uniqueness to the considered problem. The problem is formulated in the framework of multi-valued mappings. The notion of entropy measure-valued solutions is used to prove the so-called contraction principle and comparison principle.

A semigroup approach to an integro-differential equation modeling slow erosion

The paper is concerned with a scalar conservation law with nonlocal flux, providing a model for granular flow with slow erosion and deposition. While the solution u=u(t,x) can have jumps, the inverse function x=x(t,u) is always Lipschitz continuous; its derivative has bounded variation and satisfies a balance law with measure-valued sources. Using a backward Euler approximation scheme combined with a nonlinear projection operator, we construct a continuous semigroup whose trajectories are the unique entropy weak solutions to this balance law. Going back to the original variables, this yields the global well-posedness of the Cauchy problem for the granular flow model.

On the controllability of the non-isentropic 1-D Euler equation

In this paper, we examine the question of the boundary controllability of the one-dimensional non-isentropic Euler equation for compressible polytropic gas, in the context of weak entropy solutions. We consider the system in Eulerian coordinates and the one in Lagrangian coordinates. We obtain for both systems a result of controllability toward constant states (with the limitation γ<53 on the adiabatic constant for the Lagrangian system). The solutions that we obtain remain of small total variation in space if the initial condition is itself of sufficiently small total variation.

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).

On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of \({\mathbb{R}^{d}}\) and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.

Asymptotic Behavior of Global Entropy Solutions for Nonstrictly Hyperbolic Systems with Linear Damping

We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that ast→∞the entropy solutions tend to zero in theLpnorm.

HYPERBOLIC BALANCE LAWS WITH ENTROPY ON A CURVED SPACETIME: THE WEAK–STRONG UNIQUENESS THEORY

We are interested in nonlinear partial differential equations of hyperbolic type, written as first-order balance laws posed on a curved spacetime, whose unknown is typically a section of the bundle of future-oriented, timelike vectors. We are especially interested in dealing with systems and solutions with low regularity and, to this end, we introduce here the notion of "strictly convex entropy field". We then prove that a system of balance laws endowed with such an entropy admits a symmetrization which, however, may be "degenerate" if the entropy is not uniformly convex or if the constitutive laws defining the balance laws have limited regularity. Next, we establish a weak–strong uniqueness and stability theorem for the initial value problem associated with balance laws endowed with a strictly convex entropy field. We compare two solutions with limited regularity: on the one hand, a continuous solution that need not be Lipschitz continuous and, on the other hand, a weak solution that satisfies an "entropy inequality". Finally, we apply our theory to the Euler system for compressible fluid flows on a curved spacetime, and we exhibit an entropy field, which is strictly convex but fails to be uniformly convex as vacuum is approached. This leads us to a uniqueness theorem for (both the relativistic and non-relativistic versions of) the Euler system, which applies to a continuous solution with vacuum and an entropy weak solution with vacuum.

Radon measure-valued solutions for some quasilinear degenerate elliptic equations

We prove the existence of suitably defined weak Radon measure-valued solutions for a class of quasilinear elliptic equations. Moreover, we prove uniqueness results introducing the notion of “weak entropy solutions.”

Singular Limiting Induced from Continuum Solutions and the Problem of Dynamic Cavitation

In the works of Pericak-Spector and Spector (Arch Rational Mech Anal. 101:293–317, 1988, Proc. Royal Soc. Edinburgh Sect A 127:837–857, 1997) a class of self-similar solutions are constructed for the equations of radial isotropic elastodynamics that describe cavitating solutions. Cavitating solutions decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (for polyconvex energies) due to point-singularities at the cavity. To resolve this paradox, we introduce the concept of singular limiting induced from continuum solution (or slic-solution), according to which a discontinuous motion is a slic-solution if its averages form a family of smooth approximate solutions to the problem. It turns out that there is an energetic cost for creating the cavity, which is captured by the notion of slic-solution but neglected by the usual entropic weak solutions. Once this cost is accounted for, the total mechanical energy of the cavitating solution is in fact larger than that of the homogeneously deformed state. We also apply the notion of slic-solutions to a one-dimensional example describing the onset of fracture, and to gas dynamics in Langrangean coordinates with Riemann data inducing vacuum in the wave fan.

The entropy weak solution to a generalized Degasperis-Procesi equation

A nonlinear generalization of the Degasperis-Procesi equation is investigated. The well-posedness of entropy weak solutions for the Cauchy problem of the equation is established in the space L 1 (R)∩ L ∞ (R).MSC:35G25, 35L05.