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.     

Asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations

In this paper, we consider the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations. First, for 1D quasi-linear hyperbolic systems with zero eigenvalues, we establish the existence and uniqueness of semiglobal classical solution to the one-sided mixed initial-boundary value problem on a semibounded initial axis and discuss the asymptotic behavior of the corresponding solutions under different hypotheses on the initial data. Based on these results, we obtain the asymptotic stability of the exact boundary controllability of nodal profile for 1D quasi-linear wave equations on a semibounded time interval.     

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.     

Well-posedness,global existence,and blowup phenomena for a periodic quasi-linear hyperbolic equation

We establish the local well-posedness of a recently derived model for small-amplitude, shallow water waves. For a large class of initial data we prove global existence of the corresponding solution. Criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained, and an existence and uniqueness result for a class of global weak solutions is also given. © 1998 John Wiley & Sons, Inc.     

Global solutions of abstract quasi-linear evolution equations of “hyperbolic” type

The problem of existence and uniqueness of global classical solutions of abstract quasi-linear evolution equations is considered in a general Banach space. The results obtained here are applied to the initial value problems for hyperbolic partial differential equations.     

Relaxation approximation of the Euler equations

The aim of this paper is to show how solutions to the one-dimensional compressible Euler equations can be approximated by solutions to an enlarged hyperbolic system with a strong relaxation term. The enlarged hyperbolic system is linearly degenerate and is therefore suitable to build an efficient approximate Riemann solver. From a theoretical point of view, the convergence of solutions to the enlarged system towards solutions to the Euler equations is proved for local in time smooth solutions. We also show that arbitrarily large shock waves for the Euler equations admit smooth shock profiles for the enlarged relaxation system. In the end, we illustrate these results of convergence by proposing a numerical procedure to solve the enlarged hyperbolic system. We test it on various cases.     

Initial boundary value problem for the 3D quasilinear hyperbolic equations with nonlinear damping

We investigate global existence and asymptotic behavior of the 3D quasilinear hyperbolic equations with nonlinear damping on a bounded domain with slip boundary condition, which describes the propagation of heat waves for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of classical solutions are obtained when the initial data are near its equilibrium. Time asymptotically, the internal energy is conjectured to satisfy the porous medium equation and the heat flux obeys the classical Darcy’s-type law. Based on energy estimates, we show that the classical solution converges to steady state exponentially fast in time. Moreover, we also verify that the same is true for the corresponding initial boundary value problem of porous medium equation and thus justifies the validity of Darcy’s-type law in large time.     

Global Classical Solutions to the Equations of Two-dimensional MHD Transverse Flow

A new method for constructing the symmetric form for the hyperbolic systems is introduced. Then a new symmetric form of the equations of MHD transverse flow is constructed by adding an additional equation. With this new form, we obtain the local existence of smooth solution including the case that the initial density may tend to the vacuum state at infinity. Furthermore, the uniformly a priori estimation for the classical solutions is established and the global smooth solutions for a kind of initial data are obtained.     

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.     

Classical solutions of hyperbolic functional differential systems

The paper deals with a generalized Cauchy problem for quasi-linear hyperbolic functional differential systems. The unknown function is the functional variable in the system of equations and the partial derivatives appear in the classical sense. A theorem on the local existence of a solution is proved. The initial problem is transformed into a system of functional integral equations for an unknown function and for their partial derivatives with respect to spatial variables. A method of bicharacteristics and integral inequalites are applied.     

Riemann problem and elementary wave interactions in isentropic magnetogasdynamics

In this paper, we consider the Riemann problem for a quasilinear hyperbolic system of equations governing the one dimensional unsteady simple wave flow of an isentropic, inviscid and perfectly conducting compressible fluid, subjected to a transverse magnetic field. This class of equations includes, as a special case, the equations of isentropic gasdynamics. We study the shock and rarefaction waves and their properties, and discuss the geometry of shock curves using the Riemann invariant coordinates. Under certain conditions, we show the existence and uniqueness of the solution to the Riemann problem for arbitrary initial data, and then discuss the vacuum state in isentropic magnetogasdynamics. Finally, we discuss numerical results for different initial data, and discuss all possible interactions of elementary waves. It is noticed that although the magnetogasdynamic system is more complex than the corresponding gasdynamic system, all the parallel results remain identical. However, unlike the ordinary gasdynamic case, the solution inside rarefaction waves in magnetogasdynamics cannot be obtained directly and explicitly; indeed, it requires an extra iteration procedure. It is also observed that the presence of a magnetic field makes both the shock and rarefaction stronger compared to what they would have been in the absence of a magnetic field.     

On the stochastic quasi-linear symmetric hyperbolic system

We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for a quasi-linear symmetric hyperbolic system with random noise in Rd. When the noise is multiplicative satisfying some nondegenerate conditions and the initial data are sufficiently small, we show that the solution exists globally in time in probability, i.e., the probability of global existence can be made arbitrarily close to one if the initial date are small accordingly.     

Global Existence of Classical Solutions for Some Oldroyd-B Model via the Incompressible Limit

In this paper, we prove local and global existence of classical solutions for a system of equations concerning an incompressible viscoelastic fluid of Oldroyd-B type via the incompressible limit when the initial data are sufficiently small.     

Cauchy problem for quasilinear hyperbolic systems of shallow water equations

This article investigates the Cauchy problem for two different models (modified and classical), governed by quasilinear hyperbolic systems that arise in shallow water theory. Under certain reasonable hypotheses on the initial data, we obtain the global smooth solutions for both the systems. The bounds on simple wave solutions of the modified system are shown to depend on the parameter H characterizing the advective transport of impulse. Similarly the bounds on simple wave solutions of the classical system describing the flow over a sloping bottom with profile b(x) are shown to depend on the bottom topography. On the other hand, if the initial data are specified differently, then it is shown that solutions for both the systems exhibit finite time blow-up from specific smooth initial data. Moreover, we show that an increase in H and convexity of b would reduce the time taken for the solutions to blow up.     

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

We consider several nonlinear evolution equations sharing a nonlinearity of the form ?²u²/?t². Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.     

Singularités de solutions d'équations aux dérivées partielles

We develop the general mathematical setting necessary to study the singularities of local solutions of the quasi-linear first-order systems of PDEs with a free initial condition. In the good cases, it is possible to describe these singularities as a function of the free initial conditions satisfied by the solutions. Using the transversality theorems, it is then possible to describe the singularities of generic solutions, and of generic families of solutions under deformation of the initial conditions. We apply this study by giving classifications of an important classe of hyperbolic quasi-linear first-order systems in the plane, the reducible systems, and of an almost general class of hyperbolic quasi-linear second-order equations in the plane.     

Shallow water equations: viscous solutions and inviscid limit

We establish the inviscid limit of the viscous shallow water equations to the Saint-Venant system. For the viscous equations, the viscosity terms are more degenerate when the shallow water is close to the bottom, in comparison with the classical Navier-Stokes equations for barotropic gases; thus, the analysis in our earlier work for the classical Navier-Stokes equations does not apply directly, which require new estimates to deal with the additional degeneracy. We first introduce a notion of entropy solutions to the viscous shallow water equations and develop an approach to establish the global existence of such solutions and their uniform energy-type estimates with respect to the viscosity coefficient. These uniform estimates yield the existence of measure-valued solutions to the Saint-Venant system generated by the viscous solutions. Based on the uniform energy-type estimates and the features of the Saint-Venant system, we further establish that the entropy dissipation measures of the viscous solutions for weak entropy-entropy flux pairs, generated by compactly supported C ² test-functions, are confined in a compact set in H ^?1, which yields that the measure-valued solutions are confined by the Tartar-Murat commutator relation. Then, the reduction theorem established in Chen and Perepelitsa [5] for the measure-valued solutions with unbounded support leads to the convergence of the viscous solutions to a finite-energy entropy solution of the Saint-Venant system with finite-energy initial data, which is relative with respect to the different end-states of the bottom topography of the shallow water at infinity. The analysis also applies to the inviscid limit problem for the Saint-Venant system in the presence of friction.     

Introduction to nonclassical shock solutions of the modified Burgers‐Korteweg‐de Vries equation

A general property of nonlinear hyperbolic equations is the eventual formation of discontinuities in the propagating signal. These discontinuities are not uniquely defined by the initial data for the problem and a central issue is the identification of acceptable weak solutions. Particular difficulties arise when the hyperbolic system ceases to be genuinely nonlinear in some of its characteristic fields. This equates in the case of a scalar law to the lack of convexity in the flux function. Here a representative example is provided by the modified Korteweg‐de Vries‐Burgers equation which exhibits a quadratic as well as a cubic nonlinear term and arises in a variety of engineering applications including weakly nonlinear waves in fluidized beds and two‐layer fluid flows. Its solutions have the distinguishing feature to generate undercompressive or nonclassical shocks in the hyperbolic limit with dispersion and dissipation balanced. The resulting rich variety of wave phenomena: shocks which emanate rather than absorb characteristics, compound shocks and shock fan combinations, which have no counterpart in classical shock theories is discussed.     

Criteria of unique solvability of nonlocal boundary-value problem for systems of hyperbolic equations with mixed derivatives

We consider nonlocal boundary-value problem for a system of hyperbolic equations with two independent variables. We investigate questions of existence of unique classical solution to problem under consideration. In terms of initial data we propose criteria of unique solvability and suggest algorithms of finding of solutions to nonlocal boundary-value problem. As an application we give conditions of solvability of periodic boundary-value problem for a system of hyperbolic equations.     

The Stability of Large‐Amplitude Shallow Interfacial Non‐Boussinesq Flows

The system of equations describing the shallow‐water limit dynamics of the interface between two layers of immiscible fluids of different densities is formulated. The flow is bounded by horizontal top and bottom walls. The resulting equations are of mixed type: hyperbolic when the shear is weak and the behavior of the system is internal‐wave like, and elliptic for strong shear. This ellipticity, or ill‐posedness is shown to be a manifestation of large‐scale shear instability. This paper gives sharp nonlinear stability conditions for this nonlinear system of equations. For initial data that are initially hyperbolic, two different types of evolution may occur: the system may remain hyperbolic up to internal wave breaking, or it may become elliptic prior to wave breaking. Using simple waves that give a priori bounds on the solutions, we are able to characterize the condition preventing the second behavior, thus providing a long‐time well‐posedness, or nonlinear stability result. Our formulation also provides a systematic way to pass to the Boussinesq limit, whereby the density differences affect buoyancy but not momentum, and to recover the result that shear instability cannot occur from hyperbolic initial data in that case.