Recent zbMATH articles in MSC 35L60https://zbmath.org/atom/cc/35L602021-06-15T18:09:00+00:00WerkzeugEntropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra.https://zbmath.org/1460.352252021-06-15T18:09:00+00:00"Cordesse, Pierre"https://zbmath.org/authors/?q=ai:cordesse.pierre"Massot, Marc"https://zbmath.org/authors/?q=ai:massot.marcSummary: In the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and nonconservative first-order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of supplementary conservation laws, and more specifically of an entropy supplementary conservation law for smooth solutions, well known, there exists so far no general extension to obtain such supplementary conservation laws when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms with the objective of obtaining specifically an entropy supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first-order terms for the heavy species coupled to second-order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow.https://zbmath.org/1460.350682021-06-15T18:09:00+00:00"Tsuge, Naoki"https://zbmath.org/authors/?q=ai:tsuge.naokiThe auhtor studies the one-dimensional non-steady isentropic compressible Euler flow in a nozzle. The nozzle is infinitely long and described by an \(x\)-dependent cross-section function in the equations. Equations are written in terms of density and momentum. The gas is barotropic, so that pressure is a power function of density with the adiabatic exponent is from [1,5/3]. The Cauchy problem for arbitrarily large initial data is studied. The aim is to establish the global existence of the entropy solution with sonic state.
For the entire collection see [Zbl 1453.35003].
Reviewer: Ilya A. Chernov (Petrozavodsk)A posteriori error estimates for self-similar solutions to the Euler equations.https://zbmath.org/1460.352682021-06-15T18:09:00+00:00"Bressan, Alberto"https://zbmath.org/authors/?q=ai:bressan.alberto"Shen, Wen"https://zbmath.org/authors/?q=ai:shen.wenSummary: The main goal of this paper is to analyze a family of ``simplest possible'' initial data for which, as shown by numerical simulations, the incompressible Euler equations have multiple solutions. We take here a first step toward a rigorous validation of these numerical results. Namely, we consider the system of equations corresponding to a self-similar solution, restricted to a bounded domain with smooth boundary. Given an approximate solution obtained via a finite dimensional Galerkin method, we establish a posteriori error bounds on the distance between the numerical approximation and the exact solution having the same boundary data.A new numerical method for level set motion in normal direction used in optical flow estimation.https://zbmath.org/1460.650172021-06-15T18:09:00+00:00"Frolkovič, Peter"https://zbmath.org/authors/?q=ai:frolkovic.peter"Kleinová, Viera"https://zbmath.org/authors/?q=ai:kleinova.vieraSummary: We present a new numerical method for the solution of level set advection equation describing a motion in normal direction for which the speed is given by the sign function of the difference of two given functions. Taking one function as the initial condition, the solution evolves towards the second given function. One of possible applications is an optical flow estimation to find a deformation between two images in a video sequence. The new numerical method is based on a bilinear interpolation of discrete values as used for the representation of images. Under natural assumptions, it ensures a monotone decrease of the absolute difference between the numerical solution and the target function, and it handles properly the discontinuity in the speed due to the dependence on the sign function. To find the deformation between two functions (or images), the backward tracking of characteristics is used. Two numerical experiments are presented, one with an exact solution to show an experimental order of convergence and one based on two images of lungs to illustrate a possible application of the method for the optical flow estimation.Convergence rate from hyperbolic systems of balance laws to parabolic systems.https://zbmath.org/1460.350152021-06-15T18:09:00+00:00"Li, Yachun"https://zbmath.org/authors/?q=ai:li.yachun"Peng, Yue-Jun"https://zbmath.org/authors/?q=ai:peng.yuejun"Zhao, Liang"https://zbmath.org/authors/?q=ai:zhao.liang.2|zhao.liang.4|zhao.liang.1|zhao.liang.5|zhao.liang.3|zhao.liangSummary: It is proved recently that partially dissipative hyperbolic systems converge globally-in-time to parabolic systems in a slow time scaling, when initial data are smooth and sufficiently close to constant equilibrium states. Based on this result, we establish error estimates between the smooth solutions of the hyperbolic systems of balance laws and those of the parabolic limit systems in one space dimension. The proof of the error estimates uses a stream function technique together with energy estimates. As applications of the results, we give five examples arising from physical models.Riemann problem and wave interactions for a class of strictly hyperbolic systems of conservation laws.https://zbmath.org/1460.352312021-06-15T18:09:00+00:00"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3"Zhang, Yanyan"https://zbmath.org/authors/?q=ai:zhang.yanyanSummary: A class of strictly hyperbolic systems of conservation laws are proposed and studied. Firstly, the Riemann problem with initial data of two piecewise constant states is constructively solved. The solutions involving contact discontinuities and delta shock waves are obtained. The generalized Rankine-Hugoniot relation and entropy condition for the delta shock wave are clarified and the existence and uniqueness of the delta-shock solution is proved. Furthermore, the global structure of solutions with five different configurations is constructed via investigating the interactions of delta shock waves and contact discontinuities. Finally, we present a typical example to illustrate the application of the system introduced.Stability of non-constant equilibrium solutions for the full compressible Navier-Stokes-Maxwell system.https://zbmath.org/1460.352822021-06-15T18:09:00+00:00"Feng, Yue-Hong"https://zbmath.org/authors/?q=ai:feng.yuehong"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin.9|li.xin.3|li.xin.7|li.xin.10|li.xin.6|li.xin.4|li.xin.2|li.xin.1|li.xin.12|li.xin.13|li.xin.5|li.xin.15|li.xin|li.xin.11|li.xin.14"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shuSummary: In this article we consider a Cauchy problem for the full compressible Navier-Stokes-Maxwell system arising from viscosity plasmas. This system is quasilinear hyperbolic-parabolic. With the help of techniques of symmetrizers and the smallness of non-constant equilibrium solutions, we establish that global smooth solutions exist and converge to the equilibrium solution as the time approaches infinity. This result is obtained for initial data close to the steady-states. As a byproduct, we obtain the global stability of solutions near the equilibrium states for the full compressible Navier-Stokes-Poisson system in a three-dimensional torus.