Quasilinear Conservation Laws Research Articles

Hydrodynamics for Asymmetric Simple Exclusion on a Finite Segment with Glauber-Type Source

We consider an open interacting particle system on a finite lattice. The particles perform asymmetric simple exclusion and are randomly created or destroyed at all sites, with rates that grow rapidly near the boundaries. We study the hydrodynamic limit for the particle density at the hyperbolic space-time scale and obtain the entropy solution to a boundary-driven quasilinear conservation law with a source term. Different from the usual boundary conditions introduced in Bardos et al (Commun Partial Differ Equ 4(9):1017–1034, https://doi.org/10.1080/03605307908820117, 1979) and Otto (C R Acad Sci Paris 322(1):729–734, 1996), discontinuity (boundary layer) does not formulate at the boundaries due to the strong relaxation scheme.

Linear stability of shock profiles for a quasilinear Benney system in ℝ2 × ℝ+

Following Dias et al. [Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws, J. Differential Equations 251 (2007) 555–563], we study the linearized stability of a pair [Formula: see text], where [Formula: see text] is a shock profile for a family of quasilinear hyperbolic conservation laws in [Formula: see text] coupled with a semilinear Schrödinger equation.

On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law

We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law [Formula: see text] which can describe the interaction between a laser beam and a fluid flow (see [I.-S. Aranson and L. Kramer, The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys. 74 (2002) 99–143]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore, we prove the existence of standing wave solutions of the form [Formula: see text] in several cases.

On the Variational Approach to Systems of Quasilinear Conservation Laws

The paper contains results concerning the development of a new approach to the proof of existence theorems for generalized solutions to systems of quasilinear conservation laws. This approach is based on reducing the search for a generalized solution to analyzing extremal properties of a certain set of functionals and is referred to as a variational approach. The definition of a generalized solution can be naturally reformulated in terms of the existence of critical points for a set of functionals, which is convenient within the approach proposed. The variational representation of generalized solutions, which was earlier known for Hopf-type equations, is generalized to systems of quasilinear conservation laws. The extremal properties of the functionals corresponding to systems of conservation laws are described within the variational approach, and a strategy for proving the existence theorem is outlined. In conclusion, it is shown that the variational approach can be generalized to the two-dimensional case.

Systems of quasilinear conservation laws and algorithmization of variational principles

A previously formulated new approach to the consideration of systems of quasilinear hyperbolic equations on the basis of variational principles is described in more detail in the case of special systems of three equations. It is shown that each field of characteristics can be represented as a solution of a variational problem. Moreover, the Rankine–Hugoniot relations at the corner points of the characteristics or at the intersections of the characteristics of a single family hold automatically. In the simplest case of the Hopf equation, a numerical algorithm is constructed on the basis of a variational principle.

Cauchy–Gelfand problem for quasilinear conservation law

We obtain the precise asymptotic (t→∞) for solution f(x,t) of Cauchy–Gelfand problem for quasilinear conservation law ∂f∂t+φ(f)∂f∂x=0 with initial data of bounded variation f(x,0)=f0(x). The main theorem develops results of Liu (1981) [22], Kruzhkov, Petrosjan (1987) [20], Henkin, Shananin (2004) [10], Henkin (2012) [9]. Proofs are based on vanishing viscosity estimates and localized Maxwell type conservation laws. The main application consists in the reconstruction of parameters of initial data responsible for location of inviscid shock waves in the solution f(x,t).

Renormalized entropy solutions to the Cauchy problem for first order quasilinear conservation laws in the class of periodic functions

We introduce the notion of a periodic in spatial variables renormalized entropy solution to the Cauchy problem for a first order quasilinear conservation law. We prove the existence and uniqueness theorems and comparison principle. We also clarify relations with generalized entropy solutions. Bibliography: 19 titles. Illustrations: 1 figure.

Existence and uniqueness of unbounded entropy solutions of the Cauchy problem for first-order quasilinear conservation laws

We indicate conditions for the well-posedness of the Cauchy problem for a scalar quasilinear conservation law in the class of locally bounded functions. We construct examples showing that if these conditions are violated, then the Cauchy problem may fail to have a generalized entropy solution.

Uniqueness of a generalized entropy solution to the Cauchy problem for a quasilinear conservation law with convex flux

For a one-dimensional conservation law with convex flux function we prove the uniqueness of a locally bounded generalized entropy solution to the Cauchy problem with an arbitrary bounded measurable initial function. Bibliography: 12 titles. Illustrations: 2 figures.

Compact Accurately Boundary-Adjusting high-REsolution Technique for fluid dynamics

A novel high-resolution numerical method is presented for one-dimensional hyperbolic problems based on the extension of the original Upwind Leapfrog scheme to quasi-linear conservation laws. The method is second-order accurate on non-uniform grids in space and time, has a very small dispersion error and computational stencil defined within one space–time cell. For shock-capturing, the scheme is equipped with a conservative non-linear correction procedure which is directly based on the maximum principle. Plentiful numerical examples are provided for linear advection, quasi-linear scalar hyperbolic conservation laws and gas dynamics and comparisons with other computational methods in the literature are discussed.

On a class of systems of quasilinear conservation laws

We consider hyperbolic conservation laws on matrix algebras. We describe entropies of such systems and study properties of generalized entropy solutions and strong generalized entropy solutions to the Cauchy problem. Bibliography: 15 titles.

EXISTENCE OF WEAK SOLUTIONS FOR A QUASILINEAR VERSION OF BENNEY EQUATIONS

Benney introduced a general strategy for deriving systems of nonlinear partial differential equations associated with long- and short-wave solutions. The semi-linear Benney system was studied recently by Tsutsumi and Hatano. Here, we tackle the nonlinear version of it and using compensated compactness techniques, we prove the global existence of weak solutions to the Cauchy problem, in the case that the equation for the amplitude of the long wave is a quasilinear conservation law with flux f(v) = av2 - bv3 where a, b are constants with b > 0.

On the theory of isentropic solutions of quasilinear conservation laws

This paper studies isentropic solutions of quasilinear first-order equations with two independent variables and a flux function that is only continuous. The isentropic solutions are characterized by the requirement that the S. N. Kruzhkov entropy conditions hold for these solution with the equality sign. It turns out that the existence of a nonconstant isentropic solution imposes rather strong restrictions on the nonlinearity. In particular, it is shown that on the minimal interval containing the essential image of the isentropic solution, the flux function satisfies the local Lipschitz condition, and its generalized derivative is a function of locally bounded variation. Also, it is proved that when the flux function is nonlinear, any isentropic solution is continuous on nondegenerate intervals.

Isentropic solutions of quasilinear equations of the first order

Conditions for the existence of non-trivial isentropic solutions of quasilinear conservation laws are found. Applications to the problem of the functional dependence between partial derivatives of a smooth function of two variables are presented. In particular, necessary conditions on a function for the equation to have non-trivial -smooth solutions are found.

Convergence of monotone finite-difference approximations of a quasilinear conservation law

We consider a class of finite-difference approximations of the weak solution of the Cauchy problem including both explicit and implicit schemes. Monotonicity conditions are stated. Stability and convergence are investigated, accuracy bounds are obtained.

Stability and Convergence of Monotone Finite-Difference Approximations to a Quasilinear Conservation Law

for all k ∈ R. Here Φ(v, k) = sgn(v − k) [fs(v)− fs(k)]. Monotone schemes approximating Eq. (1) were studied in [3–9]. In the present paper, we generalize results similar to those obtained in [3–9] to a wider class of monotone schemes including implicit as well as explicit finite-difference approximations. Let tn, xi be the nodes of a rectangular grid. In the half-space { x ∈ R, t ≥ 0 } , we consider a grid function uτ that takes constant values ui [n = 0, 1, . . . , i = (i1, i2, . . . , id), is = 0,±1, . . .] in the parallelepipeds Πi = {tn ≤ t < tn+1}×Πi, where Πi = {xis ≤ xs < xis + his} and his = xis+1−xis . Next, we introduce the sets K(A) = { v | v ∈ L1 ( R ) , |v| ≤ A } and Kτ (A) = PτK(A), where Pτ is the operator given by the formula

Weak solutions for equations defined by accretive operators II: relaxation limits

In the first part of this paper we define solutions for certain nonlinear equations defined by accretive operators, “dissipative solution”. This kind of solution is equivalent to the viscosity solutions for Hamilton–Jacobi equations and to the entropy solutions for conservation laws. In this paper we use dissipative solutions to obtain several relaxation limits for systems of semilinear transport equations and quasilinear conservation laws. These converge to diffusion second-order equations and in one case to a single conservation law. The relaxation limit is obtained using a version of the perturbed test function method to pass to the limit. This guarantees existence for the considered equations.

Convergence of a splitting scheme applied to the Ruijgrok–Wu model of the Boltzmann equation

This paper deals with upwind splitting schemes for the Ruijgrok-Wu model (Physica A 113 (1982) 401–416) of the kinetic theory of rarefied gases in the fluid-dynamic scaling. We prove the stability and the convergence for these schemes. The relaxation limit is also investigated and the limit equation is proved to be a first-order quasi-linear conservation law. The loss of quasi-monotonicity of the present model makes it necessary to give a more careful analysis of its structure. We also obtain global error estimates in the spaces W s, p for −1⩽s⩽1/p, 1⩽p⩽∞ and pointwise error estimates for the approximate solution. The proof naturally uses the framework introduced by Nessyahu and Tadmor (SIAM J. Numer Anal. 29 (1992) 1505–1519) due to the convexity of the flux function.

Scaling limits for the Ruijgrok–Wu model of the Boltzmann equation

AbstractThe Ruijgrok–Wu model of the kinetic theory of rarefied gases is investigated both in the fluid‐dynamic and hydro‐dynamic scalings. It is shown that the first limit equation is a first order quasilinear conservation law, whereas the limit equation in the diffusive scaling is the viscous Burgers equation. The main difficulties came from initial layers that we handle here. Copyright © 2001 John Wiley &amp; Sons, Ltd.

A Godunov-type scheme for relativistic magnetohydrodynamics

The basic properties of relativistic magnetohydrodynamics, as a hyperbolic system of quasi-linear conservation laws, are discussed. These are then used to develop a multidimensional Godunov-type numerical scheme that enforces the magnetic flux conservation. This scheme is based on linear Riemann solvers and has second-order accuracy in smooth regions. The results of thorough test calculations demonstrate that the scheme is robust and can cope with truly ultrarelativistic problems.