Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero. The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248. The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.     

Numerical Approximation of Oscillatory Solutions of Hyperbolic-Elliptic Systems of Conservation Laws by Multiresolution Schemes

The generic structure of solutions of initial value problems of hyperbolic-elliptic systems, also called mixed systems, of conservation laws is not yet fully understood. One reason for the absence of a core well-posedness theory for these equations is the sensitivity of their solutions to the structure of a parabolic regularization when attempting to single out an admissible solution by the vanishing viscosity approach. There is, however, theoretical and numerical evidence for the appearance of solutions that exhibit persistent oscillations, so-called oscillatory waves, which are (in general, measure-valued) solutions that emerge from Riemann data or slightly perturbed constant data chosen from the interior of the elliptic region. To capture these solutions, usually a fine computational grid is required. In this work, a version of the multiresolution method applied to a WENO scheme for systems of conservation laws is proposed as a simulation tool for the efficient computation of solutions of oscillatory wave type. The hyperbolic-elliptic $2 \times 2$ systems of conservation laws considered are a prototype system for three-phase flow in porous media and a system modeling the separation of a heavy-buoyant bidisperse suspension. In the latter case, varying one scalar parameter produces elliptic regions of different shapes and numbers of points of tangency with the borders of the phase space, giving rise to different kinds of oscillation waves.     

Stability of shock waves for the Broadwell equations

For the Broadwell model of the nonlinear Boltzmann equation, there are shock profile solutions, i.e. smooth traveling waves that connect two equilibrium states. For weak shock waves, we prove asymptotic (in time) stability with respect to small perturbations of the initial data. Following the work of Liu [7] on shock wave stability for viscous conservation laws, the method consists of analyzing the solution as the sum of a shock wave, a diffusive wave, a linear hyperbolic wave and an error term. The diffusive and linear hyperbolic waves are approximate solutions of the fluid dynamic equations corresponding to the Broadwell model. The error term is estimated using a variation of the energy estimates of Kawashima and Matsumura [6] and the characteristic energy method of Liu [7].Research supported by the Office of Naval Research through grant N00014-81-0002 and by the National Science Foundation through grant NSF-MCS-83-01260Research supported by the National Science Foundation through grant DMS-84-01355     

The validity of nonlinear geometric optics for weak solutions of conservation laws

The method of weakly nonlinear geometric optics is one of the main formal perturbation techniques used in analyzing nonlinear wave motion for hyperbolic systems. The tacit assumption in using such perturbation methods is that the corresponding solutions of the hyperbolic system remain smooth; since shock waves typically form in such solutions, these assumptions are rarely satisfied in practice. Nevertheless, in a variety of applied contexts, these methods give qualitatively reliable answers for discontinuous weak solutions. Here we give a rigorous proof for the validity of nonlinear geometric optics for general weak solutions of systems of hyperbolic conservation laws in a single space variable. The methods of proof do not mimic the formal construction of weakly nonlinear asymptotics but instead rely on structural symmetries of the approximating equations, stability estimates for intermediate asymptotic times, and the rapid decay in variation of weak solutions for large asymptotic times.Partially supported by NSF Grant No. DMS-8301135Partially supported by NSF Grant No. MCS-81-02360 and ARO Grant No. 483964-25530     

Generalized variational principles,global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics

We study systems of conservation laws arising in two models of adhesion particle dynamics. The first is the system of free particles which stick under collision. The second is a system of gravitationally interacting particles which also stick under collision. In both cases, mass and momentum are conserved at the collisions, so the dynamics is described by 2×2 systems of conservations laws. We show that for these systems, global weak solutions can be constructed explicitly using the initial data by a procedure analogous to the Lax-Oleinik variational principle for scalar conservation laws. However, this weak solution is not unique among weak solutions satisfying the standard entropy condition. We also study a modified gravitational model in which, instead of momentum, some other weighted velocity is conserved at collisions. For this model, we prove both existence and uniqueness of global weak solutions. We then study the qualitative behavior of the solutions with random initial data. We show that for continuous but nowhere differentiable random initial velocities, all masses immediately concentrate on points even though they were continuously distributed initially, and the set of shock locations is dense.     

Bifurcation and Solitary Waves of the Combined KdV and mKdV Equation

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigated systematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation parameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence of breather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutions are given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials is utilized to give simpler forms for the pulse-like solitary wave solutions. In view of the references, our results are the most complete and the theoretical methods are the simplest hitherto.     

Nonclassical Lie symmetry and conservation laws of the nonlinear time-fractional Korteweg–de Vries equation

In this paper, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equations in the Riemann–Liouville concept. In this study, first, we employ the classical and nonclassical Lie symmetries(LS) to acquire similarity reductions of the nonlinear fractional far field Korteweg–de Vries(KdV)equation, and second, we find the related exact solutions for the derived generators. Finally,according to the LS generators acquired, we construct conservation laws for related classical and nonclassical vector fields of the fractional far field Kd V equation.     

Weak Solutions with Decreasing Energy¶of Incompressible Euler Equations

Weak solution of the Euler equations is an L²-vector field u(x, t), satisfying certain integral relations, which express incompressibility and the momentum balance. Our conjecture is that some weak solutions are limits of solutions of viscous and compressible fluid equations, as both viscosity and compressibility tend to zero; thus, we believe that weak solutions describe turbulent flows with very high Reynolds numbers. Every physically meaningful weak solution should have kinetic energy decreasing in time. But the existence of such weak solutions have been unclear, and should be proven. In this work an example of weak solution with decreasing energy is constructed. To do this, we use generalized flows (GF), introduced by Y. Brenier. GF is a sort of a random walk in the flow domain, such that the mean kinetic energy of particles is finite, and the particle density is constant. We construct a GF such that fluid particles collide and stick; this sticking is a sink of energy. The GF which we have constructed is a GF with local interaction; this means that there are no external forces. The second important property is that the particle velocity depends only on its current position and time; thus we have some velocity field, and we prove that this field is a weak solution with decreasing energy of the Euler equations. The GF is constructed as a limit of multiphase flows (MF) with the mass exchange between phases.     

Finite volume schemes for dispersive wave propagation and runup

Finite volume schemes are commonly used to construct approximate solutions to conservation laws. In this study we extend the framework of the finite volume methods to dispersive water wave models, in particular to Boussinesq type systems. We focus mainly on the application of the method to bidirectional nonlinear, dispersive wave propagation in one space dimension. Special emphasis is given to important nonlinear phenomena such as solitary waves interactions, dispersive shock wave formation and the runup of breaking and non-breaking long waves.     

Initial Boundary Value Problem for Conservation Laws

This paper concerns the initial boundary value problems for some systems of quasilinear hyperbolic conservation laws in the space of bounded measurable functions. The main assumption is that the system under study admits a convex entropy extension. It is proved that then any twicely differentiable entropy fluxes have traces on the boundary if the bounded solutions are generated by either Godunov schemes or by suitable viscous approximations. Furthermore, in the case that the weak interior solutions are generated by Godunov schemes, any Lipschitz continuous entropy fluxes corresponding to convex entropies have traces on the boundary and the traces are bounded above by computable numerical boundary values. This in particular gives a trace formula for the flux functions in terms of the numerical boundary data. We also investigate the formulation of boundary conditions for systems of hyperbolic conservation laws. It is shown that the set of expected boundary values derived from the viscous approximation contains the one derived in terms of the boundary Riemann problems, and the converse is not true in general. The general theory is then applied to some specific examples. First, several new facts are obtained for convex scalar conservation laws. For example, we give example which show that Godunov schemes produce numerical boundary layers. It is shown that any continuous functions of density have traces on the boundary (instead of only entropy fluxes). We also obtain interior and boundary regularity of the weak solutions for bounded measurable initial and boundary data. A generalized Oleinik entropy condition is also obtained. Next, we prove the existence of a weak solution to the initial-boundary value problem for a family of × quadratic system with a uniformly characteristic boundary condition. Received: 23 July 1996 / Accepted: 28 October 1996     

Nonlinear stability of overcompresive shock waves in a rotationally invariant system of viscous conservation laws

This paper proves that certain non-classical shock waves in a rotationally invariant system of viscous conservation laws posses nonlinear large-time stability against sufficiently small perturbations. The result applies to small intermediate magnetohydrodynamic shocks in the presence of dissipation.Research supported by Deutsche ForschungsgemeinschaftResearch supported in part by NSF Grant DMS 90-0226 and Army Grant DAAL 03-91-G-0017     

Shock profile solutions of the Boltzmann equation

Shock waves in gas dynamics can be described by the Euler Navier-Stokes, or Boltzmann equations. We prove the existence of shock profile solutions of the Boltzmann equation for shocks which are weak. The shock is written as a truncated expansion in powers of the shock strength, the first two terms of which come exactly from the Taylor tanh (x) profile for the Navier-Stokes solution. The full solution is found by a projection method like the Lyapunov-Schmidt method as a bifurcation from the constant state in which the bifurcation parameter is the difference between the speed of soundc ₀ and the shock speeds.Research supported in part by the National Science Foundation, the Army Research Office, the Air Force of Scientific Research, the Office of Naval Research, and the Department of Energy     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

On the viscous and inviscid stability of magnetohydrodynamic shock waves

We study the viscous and inviscid stability of shock waves in barotropic and full magnetohydrodynamics. We show that there are magnetohydrodynamic shock waves that are one-dimensionally stable as viscous shock profiles while they are multidimensionally strongly unstable as planar shock discontinuities.     

Shock Waves for a Discrete Velocity Gas Mixture

We introduce three new models for a binary mixture which have only 6+5, 8+5, and 12+5 velocities and study the properties of the first two. The models are plane and have five conservation laws as expected for a binary mixture in the plane case. We look for exact solutions corresponding to traveling waves, which turn out to have the properties of a structured shock wave, and study their properties. Particular attention is paid to the overshoots in the profiles of internal energy for the mixture and the two components.     

Stationary and quasi-stationary waves in even-order dissipative systems

A new equation was recently suggested by Rudenko and Robsman [1] for describing the nonlinear wave propagation in scattering media that are characterized by weak sound signal attenuation proportional to the fourth power of frequency. General self-similar properties of the solutions to this equation were studied. It was shown that stationary solutions to this equation in the form of a shock wave exhibit unusual oscillations around the shock front, as distinct from the classical Burgers equation. Here, similar solutions are studied in detail for nonlinear waves in even-order dissipative media; namely, the solutions are compared for the media with absorption proportional to the second, fourth, and sixth powers of frequency. Based on the numerical results and the self-similar properties of the solutions, the fine structure of the shock front of stationary waves is studied for different absorption laws and magnitudes. It is shown that the amplitude and number of oscillations appearing in the stationary wave profile increase with increasing power of the frequency-dependent absorption term. For initial disturbances in the form of a harmonic wave and a pulse, quasi-stationary solutions are obtained at the stage of fully developed discontinuities and the evolution of the profile and width of the shock wave front is studied. It is shown that the smoothening of the shock front in the course of wave propagation is more pronounced when the absorption law is quadratic in frequency.     

Scaling Limits of Waves in Convex Scalar Conservation Laws Under Random Initial Perturbations

We study waves in convex scalar conservation laws under noisy initial perturbations. It is known that the building blocks of these waves are shock and rarefaction waves, both are invariant under hyperbolic scaling. Noisy perturbations can generate complicated wave patterns, such as diffusion process of shock locations. However we show that under the hyperbolic scaling, the solutions converge in the sense of distribution to the unperturbed waves. In particular, randomly perturbed shock waves move at the unperturbed velocity in the scaling limit. Analysis makes use of the Hopf formula of the related Hamilton-Jacobi equation and regularity estimates of noisy processes. AMS subject classifications: 35L60, 35B40, 60H15     

Shocks and Excitation Dynamics in a Driven Diffusive Two-Channel System

We consider classical hard-core particles hopping stochastically on two parallel chains in the same or opposite directions with an inter- and intra-chain interaction. We discuss general questions concerning elementary excitations in these systems, shocks and rarefaction waves. From microscopical considerations we derive the collective velocities and shock stability conditions. The findings are confirmed by comparison to Monte Carlo data of a multi-parameter class of simple two lane driven diffusion models, which have the stationary state of a product form on a ring. Going to the hydrodynamic limit, we point out the analogy of our results to the ones known in the theory of differential equations of two conservation laws. We discuss the singularity problem and find a dissipative term that selects the physical solution.     

Exact solutions of equation of transverse vibrations for a bar with a specific cross section variation law

Flexural wave propagation along a bar whose thickness smoothly decreases down to zero within its end piece is considered. The propagation velocity tends to zero as the tapered end of the bar is approached, and the time of wave propagation to the tapered end is infinite. As a consequence, waves propagating along the bar are not reflected from the end. Previous quantitative study of the effect in the WKB approximation shows that, in the case of parabolic tapering, the WKB approximation yields a uniform asymptotics, which is valid (or invalid) for any of the bar’s cross sections. In the case of a bar with parabolic tapering, the equation of flexural vibrations of the bar has exact analytic solutions in the form of power functions. Based on these solutions, a modified WKB approximation is proposed to solve equations for bars with nonparabolic thickness variation laws. The input impedance of a bar with a parabolic tapering is calculated and analyzed.     

Multiple-mode diffusion waves for viscous nonstrictly hyperbolic conservation laws

We study the large-time behaviors of solutions of viscous conservation laws whose inviscid part is a nonstrictly hyperbolic system. The initial data considered here is a perturbation of a constant state. It is shown that the solutions converge to single-mode diffusion waves in directions of strictly hyperbolic fields, and to multiple-mode diffusion waves in directions of nonstrictly hyperbolic fields. The multiple-mode diffusion waves, which are the new elements here, are the self-similar solutions of the viscous conservation laws projected to the nonstrictly hyperbolic fields, with the nonlinear fluxes replaced by their quadratic parts. The convergence rate to these diffusion waves isO(t ^–3/4+1/2p+) inL ^p , 1p, with >0 being arbitrarily small.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38