Recent zbMATH articles in MSC 35L65https://zbmath.org/atom/cc/35L652021-06-15T18:09:00+00:00WerkzeugA fractional degenerate parabolic-hyperbolic Cauchy problem with noise.https://zbmath.org/1460.353692021-06-15T18:09:00+00:00"Bhauryal, Neeraj"https://zbmath.org/authors/?q=ai:bhauryal.neeraj"Koley, Ujjwal"https://zbmath.org/authors/?q=ai:koley.ujjwal"Vallet, Guy"https://zbmath.org/authors/?q=ai:vallet.guySummary: We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method ``à la Kružkov'' for more general equations where the noise coefficient may depend explicitly on the spatial variable.Entropy 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.Central-upwind scheme for a non-hydrostatic Saint-Venant system.https://zbmath.org/1460.651052021-06-15T18:09:00+00:00"Chertock, Alina"https://zbmath.org/authors/?q=ai:chertock.alina-e"Kurganov, Alexander"https://zbmath.org/authors/?q=ai:kurganov.alexander"Miller, Jason"https://zbmath.org/authors/?q=ai:miller.jason"Yan, Jun"https://zbmath.org/authors/?q=ai:yan.junThe authors develop a second-order central-upwind scheme for a nonhydrostatic version of the Saint-Venant system. This scheme is well-balanced and positivity preserving. The scheme is used to study ability of the non-hydrostatic Saint-Venant system to model long-time propagation and on-shore arrival of the tsunami-type waves. It is remarked that for a certain range of the dispersive coefficients, both the shape and amplitude of the waves are preserved even when the computational grid is coarse. The importance of the dispersive terms in the description of on-shore arrival is shown.
For the entire collection see [Zbl 1453.35003].
Reviewer: Abdallah Bradji (Annaba)Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas.https://zbmath.org/1460.352692021-06-15T18:09:00+00:00"Březina, Jan"https://zbmath.org/authors/?q=ai:brezina.jan"Kreml, Ondřej"https://zbmath.org/authors/?q=ai:kreml.ondrej"Mácha, Václav"https://zbmath.org/authors/?q=ai:macha.vaclavSummary: We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical \textit{BV} solution, instead a \(\delta\)-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exist infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one. In particular, not only the criterion based on comparing dissipation rates of total energy but also a stronger version based on comparing dissipation measures fails to pick the 1D solution.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.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)Stability of smooth solutions for the compressible Euler equations with time-dependent damping and one-side physical vacuum.https://zbmath.org/1460.350312021-06-15T18:09:00+00:00"Pan, Xinghong"https://zbmath.org/authors/?q=ai:pan.xinghongThe author considers the 1D compressible Euler equations with time-dependent damping and focuses on stability of the one-side vacuum solutions. The gas is isentropic and adiabatic. The domain is time-dependent, with gas expanding from the initial compact domain. The author proves the global existence and stability of smooth solutions. Actually, the solution is obtained in the explicit form and its stability is established.
Reviewer: Ilya A. Chernov (Petrozavodsk)Flux-approximation limits of solutions to the brio system with two independent parameters.https://zbmath.org/1460.352302021-06-15T18:09:00+00:00"Zhang, Yanyan"https://zbmath.org/authors/?q=ai:zhang.yanyan"Zhang, Yu"https://zbmath.org/authors/?q=ai:zhang.yu.3Summary: By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and two-rarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a perturbed simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the parameters of the perturbed Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified. Some numerical simulations presenting the formation processes of delta shock waves and vacuum states are also presented to confirm the theory analysis.SBV regularity for Burgers-Poisson equation.https://zbmath.org/1460.352282021-06-15T18:09:00+00:00"Gilmore, Steven"https://zbmath.org/authors/?q=ai:gilmore.steven"Nguyen, Khai T."https://zbmath.org/authors/?q=ai:nguyen.khai-tSummary: The SBV regularity of weak entropy solutions to the Burgers-Poisson equation for initial data in \(\mathbf{L}^1(\mathbb{R})\) is considered. We show that the derivative of a solution consists of only the absolutely continuous part and the jump part.Conservation and constitutive equations in curvilinear coordinates.https://zbmath.org/1460.352262021-06-15T18:09:00+00:00"Cossali, Gianpietro Elvio"https://zbmath.org/authors/?q=ai:cossali.gianpietro-elvio"Tonini, Simona"https://zbmath.org/authors/?q=ai:tonini.simonaSummary: The formulation of the conservation and constitutive differential equations derived in the previous chapters was obtained under the implicit assumption that the coordinate system was a Cartesian one. In practical problems it is sometime useful to switch to more natural coordinate systems, where the actual form of the differential equations may be simplified, thanks to some symmetry properties of the problem. For example, when dealing with the heating and evaporation of a spherical drop, the natural coordinate system is the spherical one, since in such a system the governing differential equations may assume a much simpler form.
For the entire collection see [Zbl 1459.94002].Nonlinear stability in three-layer channel flows.https://zbmath.org/1460.760632021-06-15T18:09:00+00:00"Papaefthymiou, E. S."https://zbmath.org/authors/?q=ai:papaefthymiou.e-s"Papageorgiou, D. T."https://zbmath.org/authors/?q=ai:papageorgiou.demetrios-tSummary: The nonlinear stability of viscous, immiscible multilayer flows in plane channels driven both by a pressure gradient and gravity is studied. Three fluid phases are present with two interfaces. Weakly nonlinear models of coupled evolution equations for the interfacial positions are derived and studied for inertialess, stably stratified flows in channels at small inclination angles. Interfacial tension is demoted and high-wavenumber stabilisation enters due to density stratification through second-order dissipation terms rather than the fourth-order ones found for strong interfacial tension. An asymptotic analysis is carried out to demonstrate how these models arise. The governing equations are \(2\times 2\) systems of second-order semi-linear parabolic partial differential equations (PDEs) that can exhibit inertialess instabilities due to interaction between the interfaces. Mathematically this takes place due to a transition of the nonlinear flux function from hyperbolic to elliptic behaviour. The concept of hyperbolic invariant regions, found in nonlinear parabolic systems, is used to analyse this inertialess mechanism and to derive a transition criterion to predict the large-time nonlinear state of the system. The criterion is shown to predict nonlinear stability or instability of flows that are stable initially, i.e. the initial nonlinear fluxes are hyperbolic. Stability requires the hyperbolicity to persist at large times, whereas instability sets in when ellipticity is encountered as the system evolves. In the former case the solution decays asymptotically to its uniform base state, while in the latter case nonlinear travelling waves can emerge that could not be predicted by a linear stability analysis. The nonlinear analysis predicts threshold initial disturbances above which instability emerges.On the Riemann problem for a hyperbolic system of temple class.https://zbmath.org/1460.352292021-06-15T18:09:00+00:00"Guerrero, Richard A. De La Cruz"https://zbmath.org/authors/?q=ai:guerrero.richard-a-de-la-cruz"Juajibioy, Juan C."https://zbmath.org/authors/?q=ai:juajibioy.juan-carlosSummary: In this chapter, we study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of temple class. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established. Moreover, we show explicitly the solution of generalized Riemann problem.
For the entire collection see [Zbl 1314.49001].Interaction of delta shock waves for a nonsymmetric Keyfitz-Kranzer system of conservation laws.https://zbmath.org/1460.352272021-06-15T18:09:00+00:00"de la Cruz, Richard"https://zbmath.org/authors/?q=ai:de-la-cruz.richard"Santos, Marcelo"https://zbmath.org/authors/?q=ai:santos.marcelo-m"Abreu, Eduardo"https://zbmath.org/authors/?q=ai:abreu.eduardoSummary: In this work, the mechanism for the formation of the delta shock wave is analyzed to deal with interaction of delta shock waves and contact discontinuities for a system of Keyfitz-Kranzer type by means of analysis and solutions of Riemann problems. A set of numerical experiments are provided, illustrating the theoretical findings numerically. A brief survey of the Keyfitz-Kranzer systems as a base model of fundamental nonlinear phenomena in applications is provided aiming to shed light on the intricate wave structure for other related models of conservation laws appearing in applied sciences.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.The global supersonic flow with vacuum state in a 2D convex duct.https://zbmath.org/1460.352722021-06-15T18:09:00+00:00"Li, Jintao"https://zbmath.org/authors/?q=ai:li.jintao"Shen, Jindou"https://zbmath.org/authors/?q=ai:shen.jindou"Xu, Gang"https://zbmath.org/authors/?q=ai:xu.gangSummary: This paper concerns the motion of the supersonic potential flow in a two-dimensional expanding duct. In the case that two Riemann invariants are both monotonically increasing along the inlet, which means the gases are spread at the inlet, we obtain the global solution by solving the problem in those inner and border regions divided by two characteristics in \((x, y)\)-plane, and the vacuum will appear in some finite place adjacent to the boundary of the duct. In addition, we point out that the vacuum here is not the so-called physical vacuum. On the other hand, for the case that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet, which means the gases have some local squeezed properties at the inlet, we show that the \(C^1\) solution to the problem will blow up at some finite location in the non-convex duct.High-order finite volume WENO schemes for non-local multi-class traffic flow models.https://zbmath.org/1460.651062021-06-15T18:09:00+00:00"Chiarello, Felisia A."https://zbmath.org/authors/?q=ai:chiarello.felisia-angela"Goatin, Paola"https://zbmath.org/authors/?q=ai:goatin.paola"Villada, Luis M."https://zbmath.org/authors/?q=ai:villada.luis-miguelSummary: This paper focuses on the numerical approximation of a class of non-local systems of conservation laws in one space dimension, arising in traffic modeling, proposed by \textit{F. A. Chiarello} and \textit{P. Goatin} [Netw. Heterog. Media 14, No. 2, 371--387 (2019; Zbl 1426.35153)]. We present the multi-class version of the Finite Volume WENO (FV-WENO) schemes \textit{C. Chalons} et al. [SIAM J. Sci. Comput. 40, No. 1, A288--A305 (2018; Zbl 1387.35406)], with quadratic polynomial reconstruction in each cell to evaluate the non-local terms in order to obtain high-order of accuracy. Simulations using FV-WENO schemes for a multi-class model for autonomous and human-driven traffic flow are presented for \(M=3\).
For the entire collection see [Zbl 1453.35003].Uniqueness of dissipative solutions to the complete Euler system.https://zbmath.org/1460.352712021-06-15T18:09:00+00:00"Ghoshal, Shyam Sundar"https://zbmath.org/authors/?q=ai:ghoshal.shyam-sundar"Jana, Animesh"https://zbmath.org/authors/?q=ai:jana.animeshSummary: Dissipative solutions have recently been studied as a generalized concept for weak solutions of the complete Euler system. Apparently, these are expectations of suitable measure valued solutions. Motivated from \textit{E. Feireisl} et al. [Commun. Partial Differ. Equations 44, No. 12, 1285--1298 (2019; Zbl 1428.35325)], we impose a one-sided Lipschitz bound on velocity component as uniqueness criteria for a weak solution in Besov space \(B^{\alpha ,\infty}_p\) with \(\alpha >1/2\). We prove that the Besov solution satisfying the above mentioned condition is unique in the class of dissipative solutions. In the later part of this article, we prove that the one sided Lipschitz condition gives uniqueness among weak solutions with the Besov regularity, \(B^{\alpha,\infty}_3\) for \(\alpha >1/3\). Our proof relies on commutator estimates for Besov functions and the relative entropy method.