Nonstrictly Hyperbolic Research Articles

Global solutions for a hyperbolic conservation system of three equations

In this short paper, we are concerned with a mathematical model, which is the variant of non-isentropic system of polytropic gas. The main contribution is to prove the a-priori L∞ estimates for the viscosity approximation solutions and to obtain a new application, on a large hyperbolic system of three equations, of the global existence framework of weak solutions in the paper “Existence of Global Entropy Solutions to a Nonstrictly Hyperbolic System” (Lu, 2005).

Exact ideal magnetohydrodynamic Riemann solutions considering the strength of intermediate shocks

Exact magnetohydrodynamic (MHD) Riemann solutions are the basis of constructing numerical schemes and benchmarks for verifying the schemes. However, non-strict hyperbolicity and nonconvexity of MHD equations contribute to the appearance of intermediate shocks, causing low efficiency of existing exact solvers and high dependence on iterative initials. Utilizing the magnetic critical Mach number proposed in this paper, all possible intermediate shocks are analyzed, parameterized, and categorized. Moreover, the possible wave structures on one side of contact discontinuity are revealed to have 25 cases, and initial conditions are classified into three categories according to the coplanar properties. Based on our findings, a new exact MHD Riemann solver is built. The robustness has been significantly improved after avoiding considerable judgments and the dependence on iterative initials. The analysis of the exact MHD Riemann solution is carried out by the characteristic properties of MHD shocks in the parameterization, and it is found that a solution space exists with the highest dimension of two dimensions under the given initial condition. It is proposed to adopt the intensities of 2 → 3 intermediate shocks as the free parameters of solution space, which can completely express the degree of solution space freedom. Finally, two examples that possess the solution space are used as verifications. The physical properties of MHD equations show that the dominant factor for the solution space is the unique characteristic property of 2 → 3 intermediate shock: the existence of an additional free parameter with tangential symmetry simultaneously.

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.

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.

Two-Dimensional (Two-Speed) Mode for a Nonstrictly Hyperbolic Problem

Two-Dimensional (Two-Speed) Mode for a Nonstrictly Hyperbolic Problem

Two-Dimensional (Two-Speed) Mode for a Nonstrictly Hyperbolic Problem

All types of solutions to the Riemann problem for a system of conservation laws that is nonstrictly hyperbolic in the sense of Petrovskii, but not in the sense of Friedrichs are described.

Exact Riemann solutions to shallow water equations

We determine completely the Riemann solutions to the shallow water equations with a bottom step, including the dry bed problem. The nonstrict hyperbolicity of this first-order system of partial differential equations leads to resonant waves and nonunique solutions. To address these difficulties we construct the L–M and R–M curves in the state space. For the bottom step elevated from left to right, we classify the L–M curve into five different cases and the R–M curve into two different cases based on the subcritical and supercritical Froude number of the Riemann initial data as well as the jump of the bottom step. The behaviors of all basic cases of the L–M and R–M curves are fully analyzed. We observe that the non–uniqueness of the Riemann solutions is due to bifurcations on the L–M or R–M curves. The possible solutions including classical waves and resonant waves as well as dry bed state are solved in a uniform framework for any given Riemann initial data.

Non-strictly Hyperbolic Systems, Singularity and Bifurcation

This paper examines the effect of non-strict hyperbolicity on singularity formation for a quasilinear 2$$\,\times \,$$×2 system. We will show that for appropriate data a singularity can form solely along a curve in the (x, t) plane where strict hyperbolicty is lost. Furthermore, this curve may bifurcate at some point. Numerous examples are considered for Riemann and smooth data.

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.

Global Existence of Solutions for a Nonstrictly Hyperbolic System

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly boundedL∞estimatesz(ρδ,ε,uδ,ε)≤B(x)andw(ρδ,ε,uδ,ε)≤βwhena(x)is increasing (similarly,w(ρδ,ε,uδ,ε)≤B(x)andz(ρδ,ε,uδ,ε)≤βwhena(x)is decreasing) for theε-viscosity andδ-flux approximation solutions of nonhomogeneous, resonant system without the restrictionz0(x)≤0orw0(x)≤0as given in Klingenberg and Lu (1997), wherezandware Riemann invariants of nonhomogeneous, resonant system;B(x)>0is a uniformly bounded function ofxdepending only on the functiona(x)given in nonhomogeneous, resonant system, andβis the bound ofB(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

Existence of Global Weak Entropy Solutions to Some Nonstrictly Hyperbolic Systems

In this paper, the Cauchy problem for nonstrictly hyperbolic systems of conservation laws modeling polymer flooding in enhanced oil recovery is studied, and a short proof of global existence of weak entropy solutions is obtained by using a new technique from the compensated compactness theorem coupled with the total variation estimate on one family of Riemann invariants. This work extends in some sense the previous works [T. Barkve, SIAM J. Appl. Math., 49 (1989), pp. 789--798; T. Johansen and R. Winther, SIAM J. Math. Anal., 19 (1988), pp. 541--566] about the global existence of weak solutions for the Riemann problem and by the author [J. Funct. Anal., 261 (2011), pp. 2797--2815] about the global weak solution of the Cauchy problem for an extended hyperbolic system of Keyfitz--Kranzer or Aw--Rascle type.

EXACT RIEMANN SOLUTIONS TO COMPRESSIBLE EULER EQUATIONS IN DUCTS WITH DISCONTINUOUS CROSS-SECTION

We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

Equations for the motion of relativistic torus in the Minkowski space R1+n

In this paper we derive the equations for the motion of relativistic torus in the Minkowski space R1+n (n⩾3). This kind of equations also describes the three-dimensional timelike extremal submanifolds in the Minkowski space R1+n. We show that these equations can be reduced to a (1+2)-dimensional quasilinear symmetric hyperbolic system and the system possesses some interesting properties such as nonstrict hyperbolicity, constant multiplicity of eigenvalues, linear degeneracy of all characteristic fields, strong null condition, etc. We also find and prove an interesting fact that all plane wave solutions to these equations are lightlike extremal submanifolds and vice versa except for a type of special solution.

Asymptotic Behavior of Global Classical Solutions of Quasilinear Non-strictly Hyperbolic Systems with Weakly Linear Degeneracy*

In this paper, we study the asymptotic behavior of global classical solutions of the Cauchy problem for general quasilinear hyperbolic systems with constant multiple and weakly linearly degenerate characteristic fields. Based on the existence of global classical solution proved by Zhou Yi et al., we show that, when t tends to infinity, the solution approaches a combination of C1 travelling wave solutions, provided that the total variation and the L1 norm of initial data are sufficiently small.

The equation for time-like extremal surfaces in Minkowski space R2+n

In this paper we investigate the equation for time-like extremal surfaces in the Minkowski space R2+n, and show that this kind of equation enjoys many interesting properties nonstrict hyperbolicity, constant multiplicity of eigenvalues, boundedness of characteristic propagation speeds, linear degeneracy of all characteristic fields, richness, etc. Without any smallness assumption of initial data, we give the necessary and sufficient condition on the global existence of classical solutions of the Cauchy problem. Based on this, we prove some global existence theorems on classical solutions of the Dirichlet problem and Neumann problem for this kind of equation. Finally, we present an explicit exact representation, involving two independent arbitrary functions of general solution.

Nonlocal and Initial Problems for Quasilinear, Nonstrictly Hyperbolic Equations with General Solutions Represented by Superposition of Arbitrary Functions

Abstract We have selected a class of hyperbolic quasilinear equations of second order, admitting parabolic degeneracy by the following criterion: they have a general solution represented by superposition of two arbitrary functions. For equations of this class we consider the initial Cauchy problem and nonlocal characteristic problems for which sufficient conditions are established for the solution solvability and uniquness; the domains of solution definition are described.

Convergence of Approximated Solutions to a Nonstrictly Hyperbolic System

Abstract In this paper, four classes of special entropy-entropy flux pairs of Lax type for the nonstrictly hyperbolic system of type (1.1) (Le Roux system) are constructed based on the solutions of the standard Fuchsion equation. The second derivatives of these entropies are all singular at the point (0,O). A careful computation for these entropies at the singular point shows the compactness of η(u1, v1)t + q(u1, v1)x in H-1 loc:(R x R+) with respect to the approximated solutions constructed by using viscosity method or Friedrichs-Lax scheme method. These entropies provide a convergence theorem in the strong topology for the artifical viscosity method or Friedrichs-Lax scheme method when applied to the Cauchy problem (1.1),(1.4) and used together with the theory of compensated compactness.

The Solution of Nonstrictly Hyperbolic Conservation Laws May Be Hard to Compute

Some simple examples of nonstrictly hyperbolic conservation laws are presented that admit certain instability properties with respect to $L^1 $-perturbations of the initial data. One of the examples shows that a piecewise constant approximation generates an approximate solution with erroneous qualitative behaviour; another example shows that the approximate solution is not well defined. For both of these examples, a classical, stable entropy solution can be given by an explicit formula. These properties may cause difficulties for numerical methods based on the solution of Riemann problems.

The Solution of the Riemann Problem for a Hyperbolic System of Conservation Laws Modeling Polymer Flooding

The global Riemann problem for a nonstrictly hyperbolic system of conservation laws modeling polymer flooding is solved. In particular, the system contains a term that models adsorption effects.

Propagation of Singularities for Nonstrictly Hyperbolic Semilinear Systems in One Space Dimension

Propagation of Singularities for Nonstrictly Hyperbolic Semilinear Systems in One Space Dimension