Non-strictly Hyperbolic System Research Articles

The Riemann problem for the main model cases of the Euler-Poisson equations

In this paper, we construct a solution to the Riemann problem for an inhomogeneous nonstrictly hyperbolic system of two equations, which is a corollary of the Euler-Poisson equations without pressure [9]. These equations can be considered for the cases of attractive and repulsive forces as well as for the cases of zero and nonzero underlying density background. The solution to the Riemann problem for each case is nonstandard and contains a delta-shaped singularity in the density component. In [16], solutions were constructed for the combination corresponding to the cold plasma model (repulsive force and nonzero background density). In this paper, we consider the three remaining cases.

The mathematical model and analysis of the nanoparticle-stabilized foam displacement

This work proposes a mathematical model to study the foam displacement in porous media stabilized by nanoparticles. We consider a simplification of the Stochastic Bubble Population balance model in local equilibrium, with nanoparticle dependence inspired by the experimental data from the literature. It consists of a non-strictly hyperbolic system of conservation laws, which is solved for the generic initial and injection conditions. We investigate the existence of a global solution as a sequence of waves following the Conservation Laws Theory. When the solution is composed of two or more waves, we present necessary and sufficient conditions to guarantee the compatibility of these wave sequences. The analytical solution for the nanoparticle-stabilized foam displacement in porous media allowed us to quantify the effect of nanoparticles on foam displacement, focusing on the breakthrough time and cumulative water production. In agreement with the literature, when only gas is injected, the breakthrough time and the water production increase with the nanoparticle concentration. Although, we also observe that the effect of nanoparticles is less pronounced for high nanoparticle concentration. Counterintuitively, during gas-water co-injection for a certain parameter range, adding nanoparticles changes the mathematical solution qualitatively, yielding a negligible effect on water production. We discuss the most favorable conditions to observe the action of nanoparticles in laboratory experiments.

Low regularity ill-posedness for elastic waves driven by shock formation

abstract: In this paper, we construct counterexamples to the local existence of low-regularity solutions to elastic wave equations in three spatial dimensions (3D). Inspired by the recent works of Christodoulou, we generalize Lindblad's classic results on the scalar wave equation by showing that the Cauchy problem for 3D elastic waves, a physical system with multiple wave-speeds, are ill-posed in $H^3(\Bbb{R}^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties of the 3D case come from the multiple wave-speeds and its associated non-strict hyperbolicity. We obtain the desired results by designing and combining a geometric approach and an algebraic approach, equipped with detailed studies and calculations of the structures and coefficients of the corresponding non-strictly hyperbolic system. Moreover, the ill-posedness we depict also applies to 2D elastic waves, which corresponds to a strictly hyperbolic case.

Entropy solutions to a non-conservative and non-strictly hyperbolic diagonal system inspired by dislocation dynamics

Entropy solutions to a non-conservative and non-strictly hyperbolic diagonal system inspired by dislocation dynamics

Delta wave interactions in a non-strictly hyperbolic system with non-convex flux

Delta wave interactions in a non-strictly hyperbolic system with non-convex flux

Stability of Riemann solutions to a class of non-strictly hyperbolic systems of conservation laws

This paper is concerned with Riemann problem for a class of non-strictly hyperbolic systems of conservation laws. The two kinds of Riemann solutions including vacuum and delta shock wave are constructed. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of delta shock wave solutions. Furthermore, by studying the interactions among of the delta shock wave and vacuum as well as contact discontinuity, the Riemann solutions with four kinds of different structures are obtained. Additionally, the stability of the Riemann solutions is obtained under certain perturbation of the initial data.

Explicit Solution of a Non-strictly Hyperbolic System with Discontinuous Flux Using a Scaling Argument

Explicit Solution of a Non-strictly Hyperbolic System with Discontinuous Flux Using a Scaling Argument

Low regularity ill-posedness for non-strictly hyperbolic systems in three dimensions

In this paper, we survey a new approach combining algebraic and geometric ideas, with which we prove low regularity ill-posedness for quasilinear hyperbolic systems with non-strict hyperbolicity in three dimensions. These systems are also associated with multiple wave-speeds.

Riemann Problem for a Class of Non-strictly Hyperbolic Systems of Conservation Laws

Riemann Problem for a Class of Non-strictly Hyperbolic Systems of Conservation Laws

Global entropy solutions for systems modelling polymer flooding in enhanced oil recovery

In this paper, we obtain the existence of global entropy solutions for the Cauchy problem of the nonstrictly hyperbolic systems modelling polymer flooding in enhanced oil recovery, under a more flexible condition on the function β(T), which models the adsorption of the polymer on rock. This work improves the previous results in the paper Lu (2013), where β(T) is limited to meas{T:β′′(T)=0}=0 or β(T)=bT for a nonnegative constant b.

Riemann problem for a non-strictly hyperbolic system in chemotaxis

<p style='text-indent:20px;'>The Riemann problem is solved for a system arising in chemotaxis. The system is of mixed-type and transitions from a hyperbolic to an elliptic region. It is genuinely nonlinear in the <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> plane except on the <inline-formula><tex-math id="M3">\begin{document}$ v $\end{document}</tex-math></inline-formula>-axis, where it is linearly degenerate. We have solved the Riemann problem in the physically relevant region up to the boundary of the hyperbolic-elliptic region, which is non-strictly hyperbolic. We also solved the problem on the linearly degenerate region. While solving the Riemann problem, we found classical shock and rarefaction waves in the hyperbolic region and contact discontinuities in the linearly degenerate region.</p>

Solution of a transport equation with discontinuous coefficients

Abstract In this article, we study initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in { - 1 , 0 , 1 } {\{-1,0,1\}} . We construct an explicit formula for the weak solution. We also study the interaction of waves and the large time asymptotic behavior of a solution for the case when the initial data is periodic with zero mean over the period and also for the case when the initial data has compact support.

Resonance in rarefaction and shock curves: Local analysis and numerics of the continuation method

We describe certain crucial steps in the development of an algorithm for finding the Riemann solution to systems of conservation laws. We relax the classical hypotheses of strict hyperbolicity and genuine nonlinearity due to Lax. First, we present a procedure for continuing wave curves beyond points where characteristic speeds coincide, i.e. at wave curve points of maximal co-dimensionality. This procedure requires strict hyperbolicity on both sides of the coincidence locus. Loss of strict hyperbolicity is regularized by means of a Generalized Jordan Chain, which serves to construct a four-fold sub-manifold structure on which wave curves can be continued. Second, we analyze the loss of genuine nonlinearity. We prove a new result: the existence of composite wave curves when the composite wave traverses either the inflection locus or an anomalous part of the non-local composite wave curve. In this sense, we find conditions under which the composite field is well defined and its singularities can be removed, allowing use of our continuation method. Finally, we present numerical examples for a non-strictly hyperbolic system of conservation laws.

Vanishing viscosity limit for a system of balance laws with general type initial data arising from 1D Saint-Venant model

We study the vanishing viscosity limit of a non-strictly hyperbolic system of balance laws known as the 1D Saint-Venant model whose solution admits δ-waves. The balance term depends on both the variables x and t and is unbounded in the space variable x unlike the work of Olenik [Usp Math. Nauk. 12(3), 3–73 (1957) (in Russian)]. We also studied the large time behavior when the viscosity parameter is positive. When the balance term depends only on the time variable, we obtained an explicit formula for the solution.

Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

Abstract This paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

Limiting behavior of scaled general Euler equations of compressible fluid flow

The aim of this article is to study the limiting behavior of the solutions for the scaled generalized Euler equations of compressible fluid flow. When the initial data is of Riemann type, we showed the existence of solution which consists of shock waves and rarefaction waves and that the distributional limit of the solutions for this system converges to the solution of a non-strictly hyperbolic system, called one dimensional model for large scale structure formation of universe as the scaling parameter vanishes. An explicit entropy and entropy flux pair are also constructed for the particular flux function (Brio system) and it is shown that the solution constructed is entropy admissible. This is a continuation of our work[23].

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

We study the interactions between classical elementary waves and delta shock wave in quasilinear hyperbolic system of conservation laws. This governing system describes a thin film of a perfectly soluble anti-surfactant solution in the limit of large capillary and P$ \acute{e} $clet numbers. This system is one of the example of non-strictly hyperbolic system whose Riemann solution consists of delta shock wave as well as classical elementary waves such as shock waves, rarefaction waves and contact discontinuities. The global structure of the perturbed Riemann solutions are constructed and analyzed case by case when delta shock wave is involved.

Dependence on the Background Viscosity of Solutions to a Prototypical NonStrictly Hyperbolic System of Conservation Laws

This paper studies a nonstrictly hyperbolic system of conservation laws in one space dimension which has been proposed in the literature as mathematically natural and as a model of magnetohydrodynamic (MHD) turbulence. It is shown that the associated Riemann problem possesses a one-parameter family of different solution operators, each of which corresponds to a different ratio of two dissipation parameters that in the MHD interpretation correspond to coefficients of fluid viscosity and electrical resistivity. (Because of a production error, this article has been updated.)

INTERACTIONS OF DELTA SHOCK WAVES FOR A CLASS OF NONSTRICTLY HYPERBOLIC SYSTEM OF CONSERVATION LAWS

In this paper, we study the perturbed Riemann problem for a class of nonstrictly hyperbolic system of conservation laws, and focuse on the interactions of delta shock waves with the shock waves and the rarefaction waves. The global solutions are constructed completely with the method of splitting delta function. In solutions, we find a new kind of nonclassical wave, which is called delta contact discontinuity with Dirac delta function in both components. It is quite different from the previous ones on which only one state variable contains the Dirac delta function. Moreover, by letting perturbed parameter " tend to zero, we analyze the stability of Riemann solutions.

Stability of stationary solutions of singular systems of balance laws

The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.