Resonantly Interacting Weakly Nonlinear Hyperbolic Waves in the Presence of Shocks: A Single Space Variable in a Homogeneous,Time Independent Medium

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves for a single space variable in a homogeneous, time independent medium. This theory extends the results previously presented by A. Majda and R. Rosales, under similar hypotheses, to the case where waves break and shocks form. Similarly the theory of nonresonant interacting waves for general hyperbolic systems developed recently by J. Hunter and J. B. Keller, when specialized to a single space variable, is included as a special case. However, we are also able to treat the general situation when resonances occur in the hyperbolic system. Such resonances are the typical case when the hyperbolic system has at least three equations and when, for example, small-amplitude periodic initial data are prescribed. In the important physical example of the 3 ? 3 system describing compressible fluid flow in a single space variable, the resonant asymptotic theory developed by the authors yields, as limit equations, a pair of inviscid Burgers equations coupled through a linear integral operator with known kernel defined through the initial data for the entropy wave. These are the same equations derived by Majda and Rosales previously. However, the waves are displaced relative to the positions prescribed by them.     

Resonantly Interacting,Weakly Nonlinear Hyperbolic Waves. II. Several Space Variables

A uniformly valid asymptotic theory of resonantly interacting high-frequency waves for nonlinear hyperbolic systems in several space dimensions is developed. When applied to the equations of two-dimensional compressible fluid flow, this theory both predicts the geometric location of the new sound waves produced from the resonant interaction of sound waves and vorticity waves as well as yielding a simplified system which governs the evolution of the amplitudes. In this important special case, this system is two Burgers equations coupled by a linear integral operator with known kernel given by the vortex strength of the shear wave. Several inherently multidimensional assumptions for the phases are needed in this theory, and theoretical examples are given which delineate these assumptions. Furthermore, explicit necessary and sufficient conditions for the validity of the earlier noninteracting wave theory of Hunter and Keller are derived; these explicit conditions indicate that generally waves resonate and interact in several dimensions.     

A Canonical System of lntegrodifferential Equations Arising in Resonant Nonlinear Acoustics

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.     

Averaging over fast gravity waves for geophysical flows with arbitary

Here a mathematically rigorous framework is developed for deriving new reduced simplified dynamical equations for geophysical flows with arbitrary potential vorticity interacting with fast gravity waves. The examples include the rotating Boussinesq and rotating shallow water equations in the quasigeostrophic limit with vanishing Rossby number. For the spatial periodic case the theory implies that the quasi—geostrophic equations are valid limiting equations in the weak topology for arbitrary initial data. Furthermore, simplified reduced equations are developed for the fashion in which the vortical waves influence the gravity waves through averaging over specific gravity wave/vortical resonances.     

Stability of the Riemann solutions for a nonstrictly hyperbolic system of conservation laws

We prove that the Riemann solutions are stable for a nonstrictly hyperbolic system of conservation laws under local small perturbations of the Riemann initial data. The proof is based on the detailed analysis of the interactions of delta shock waves with shock waves and rarefaction waves. During the interaction process of the delta shock wave with the rarefaction wave, a new kind of nonclassical wave, namely a delta contact discontinuity, is discovered here, which is a Dirac delta function supported on a contact discontinuity and has already appeared in the interaction process for the magnetohydrodynamics equations [M. Nedeljkov and M. Oberguggenberger, Interactions of delta shock waves in a strictly hyperbolic system of conservation laws, J. Math. Anal. Appl. 344 (2008) 1143-1157]. Moreover, the global structures and large time asymptotic behaviors of the solutions are constructed and analyzed case by case.     

Shock waves in the nonisentropic gas flow

In this paper we propose an extended entropy condition for general systems of hyperbolic conservation laws with several space variables. This entropy condition generalizes the well-known condition (E) of Volpert for a single conservation law with several space variables and reduces to the entropy condition proposed earlier by the author for systems with one space variable. The Riemann problem for general nonisentropic gas equations has a unique solution for initial data with arbitrarily large jumps. The occurrence of a vacuum region is observed. The projections of shock curves on the pressure-velocity plane are analyzed so as to study the interaction of weak shocks. Our results differ markedly from those of previous works in that we do not assume the equation of state to be polytropic. In fact our assumptions on the equation of state allow the pressure to be a nonconvex function of specific volume.The Riemann problem for this general system of gas equations was also treated by B. Wendroff when the initial data are near constant.     

Surface waves supported by thin-film/substrate interactions

A systematic approximation to the linear equations for small-amplitudesurface waves in an elastic half space, interacting with a residuallystressed thin film of different material bonded to its planeboundary, is developed in powers of the film thickness, assumingthe latter to be small compared to the wavelength of the disturbance.The theory is illustrated by calculating asymptotic expansionsof the wave speeds for Love and Rayleigh waves valid to secondorder in the dimensionless film thickness for a transverselyisotropic film bonded to an isotropic substrate.     

Low-Frequency Multidimensional Instabilities for Reacting Shock Waves

This paper presents a weakly nonlinear analysis for one scenario for the development of transversal instabilities in detonation waves in two space dimensions. The theory proposed and developed here is most appropriate for understanding the behavior of regular and chaotically irregular pulsation instabilities that occur in detonation fronts in condensed phases and occasionally in gases. The theory involves low-frequency instabilities and through suitable asymptotics yields a complex Ginzburg-Landau equation that describes simultaneously the evolution of the detonation front and the nonlinear interactions behind this front. The asymptotic theory mimics the familiar theory of nonlinear hydrodynamic instability in outline; however, there are several novel technical aspects in the derivation because the phenomena studied here involve a complex free boundary problem for a system of nonlinear hyperbolic equations with source terms.     

Convergence of Compressible Euler-Maxwell Equations to Compressible Euler-Poisson Equations

In this paper,the convergence of time-dependent Euler-Maxwell equations to compressible Euler-Poisson equations in a torus via the non-relativistic limit is studied. The local existence of smooth solutions to both systems is proved by using energy esti- mates for first order symmetrizable hyperbolic systems.For well prepared initial data the convergence of solutions is rigorously justified by an analysis of asymptotic expansions up to any order.The authors perform also an initial layer analysis for general initial data and prove the convergence of asymptotic expansions up to first order.     

Instability of power-law fluid flows down an incline subjected to wind stress

In this paper we investigate the effect of a prescribed superficial shear stress on the generation and structure of roll waves developing from infinitesimal disturbances on the surface of a power-law fluid layer flowing down an incline. The unsteady equations of motion are depth integrated according to the von Kármán momentum integral method to obtain a non-homogeneous system of nonlinear hyperbolic conservation laws governing the average flow rate and the thickness of the fluid layer. By conducting a linear stability analysis we obtain an analytical formula for the critical conditions for the onset of instability of a uniform and steady flow in terms of the prescribed surface shear stress. A nonlinear analysis is performed by numerically calculating the nonlinear evolution of a perturbed flow. The calculation is carried out using a high-resolution finite volume scheme. The source term is handled by implementing the quasi-steady wave propagation algorithm. Conclusions are drawn regarding the effect of the applied surface shear stress parameter and flow conditions on the development and characteristics of the roll waves arising from the instability. For a Newtonian flow subjected to a prescribed superficial shear stress, using an analytical theory, we show that the nonlinear governing equations do not admit roll waves solutions under conditions when the uniform and steady flow is linearly stable. For the case of a general power-law fluid flow with zero shear stress applied at the surface, the analytical investigation leads to a procedure for calculating the characteristics of a roll waves flow. These results are compared with those yielded by the numerical procedure.     

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

Interaction of Shallow Water Waves

In this paper, we consider the Riemann problem and interaction of elementary waves for a nonlinear hyperbolic system of conservation laws that arises in shallow water theory. This class of equations includes as a special case the equations of classical shallow water equations. We study the bore and dilatation waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem. Towards the end, we discuss numerical results for different initial data along with all possible interactions of elementary waves. It is noticed that in contrast to the p -system, the Riemann problem is solvable for arbitrary initial data, and its solution does not contain vacuum state.     

Two-Wave Interactions for Hyperbolic Signaling Problems and the Application to Supersonic Gas Dynamics

The signaling problem for a system of conservation laws in a single space variable is treated through the deployment of a perturbation analysis. Our method of approach involves the direct use of two nonlinear phase variables making possible the study of weakly nonlinear interacting waves arising from a boundary disturbance consisting of two wave modes. As a result of our analysis, the asymptotic solution is derived, and the class of admissible boundary disturbances is distinguished as well. An application is then made to gas dynamics in one space dimension to investigate the propagation and interaction of two sound waves for which the base state is taken to be a steady supersonic flow.     

Short-wave asymptotics of solutions of the Cauchy problem for hyperbolic systems with constant coefficients and with characteristics of variable multiplicity

We construct asymptotic expansions of solutions of the Cauchy problem with rapidly oscillating initial data for hyperbolic systems with constant coefficients and with characteristics of a variable multiplicity. By way of example, we consider the system of Maxwell equations.     

Superconvergence of Finite Element Approximations to Parabolic and Hyperbolic Integro-Differential Equations

1 IntroductionRecently, considerable attention has been devoted to the finite element analysis fOr par-tiaJ illtegro-differential equations, see, fOr example, Yanik and Ftweatherl1], Cannon andLin[2'3] 5 Chen and Shih[4], Lin, Thomee and Wahlbi.I5l ? Thomee and Zhang[6] and ZhangI7j8J.The main tool used for this kind of equations is the mtz-Voterra projection [5'7] as againstttitz projection fOr parabolic equation. In this paPer, we are concerned primarily with theanalysis of knot supe…     

Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids

Many interesting problems in classical physics involve the limiting behavior of quasilinear hyperbolic systems as certain coefficients become infinite. Using classical methods, the authors develop a general theory of such problems. This theory is broad enough to study a wide variety of interesting singular limits in compressible fluid flow and magneto-fluid dynamics including new constructive local existence theorems for the time-singular limit equations. In particular, the authors give an entirely self-contained classical proof of the convergence of solutions of the compressible fluid equations to their incompressible limits as the Mach number becomes small. The theory depends upon a balance between certain inherently nonlinear structural conditions on the matrix coefficients of the system together with appropriate initialization procedures. Similar results are developed also for the compressible and incompressible Navier-Stokes equations with periodic initial data independent of the viscosity coefficients as they tend to zero.     

多变量双曲方程组初值问题一类变系数差分格式的稳定性

熟知,P.D.Lax和H.O.Kreiss)对一类对称双曲方程组的“耗散型”差分格式的稳定性得到了比较完善的结果。但由于条件稍严,实际应用受到限制。朱幼兰等对一个空间变量情形,取消对称和耗散的限制,建立了较广的一类差分格式的稳定性判别准则,并对大部分常用格式给出了与常系数情形相当的稳定性条件。本文把[3]中这一主要结果推广     

Curvature Flow of Complete Convex Hypersurfaces in Hyperbolic Space

We investigate the existence, convergence, and uniqueness of modified general curvature flow (MGCF) of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.     

Sommerfeld-type conditions and uniqueness of the solution of exterior problems of the theory of elastic vibrations of aneotropic media : PMM vol. 43, no. 6, 1979, pp. 1102–1110

Steady vibrations of unbounded elastic anisotropic media are considered in three dimensions in the general case of 21 elastic constants. The problem is posed of extracting a single solution, defined in the whole space, of a system of elliptic equations for the stationary part of the displacement. The asymptotic of the fundamental solution is investigated at infinity and radiation conditions are determined which go directly over into the Sommerfeld conditions when going over to isotropic media. A uniqueness theorem is formulated for the inhomogeneous problem in unbounded domains.