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.     

Diffusive-Dispersive Traveling Waves and Kinetic Relations: Part I: Nonconvex Hyperbolic Conservation Laws

Motivated by the theory of phase transition dynamics, we consider one-dimensional, nonlinear hyperbolic conservation laws with nonconvex flux-function containing vanishing nonlinear diffusive-dispersive terms. Searching for traveling wave solutions, we establish general results of existence, uniqueness, monotonicity, and asymptotic behavior. In particular, we investigate the properties of the traveling waves in the limits of dominant diffusion, dominant dispersion, and asymptotically small or large shock strength. As the diffusion and dispersion parameters tend to 0, the traveling waves converge to shock wave solutions of the conservation law, which either satisfy the classical Oleinik entropy criterion or are nonclassical undercompressive shocks violating it.     

Pointwise Green's Function Estimates Toward Stability for Degenerate Viscous Shock Waves

ABSTRACT

We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic endstates are sonic to the hyperbolic system (the shock speed is equal to one of the characteristic speeds). In particular, we develop detailed pointwise estimates on the Green's function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic (nonintegrable) convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the Lax case and that of the undercompressive case.     

Parabolic approximations of diffusive–dispersive equations

We consider a lower-order approximation for a third-order diffusive–dispersive conservation law with nonlinear flux. It consists of a system of two second-order parabolic equations; a coupling parameter is also added. If the flux has an inflection point it is well-known, on the one hand, that the diffusive–dispersive law admits traveling-wave solutions whose end states are also connected by undercompressive shock waves of the underlying hyperbolic conservation law. On the other hand, if the diffusive–dispersive regularization vanishes, the solutions of the corresponding initial-value problem converge to a weak solution of the hyperbolic conservation law. We show that both of these properties also hold for the lower-order approximation. Furthermore, when the coupling parameter tends to infinity, we prove that solutions of initial value problems for the approximation converge to a weak solution of the diffusive–dispersive law. The proofs rely on new a priori energy estimates for higher-order derivatives and the technique of compensated compactness.     

Singular limits for a parabolic–elliptic regularization of scalar conservation laws

We consider scalar hyperbolic conservation laws with a nonconvex flux, in one space dimension. Then, weak solutions of the associated initial value problems can contain undercompressive shock waves. We regularize the hyperbolic equation by a parabolic–elliptic system that produces undercompressive waves in the hyperbolic limit regime. Moreover we show that in another limit regime, called capillarity limit, we recover solutions of a diffusive–dispersive regularization, which is the standard regularization used to approximate undercompressive waves. In fact the new parabolic–elliptic system can be understood as a low-order approximation of the third-order diffusive–dispersive regularization, thus sharing some similarities with the relaxation approximations. A study of the traveling waves for the parabolic–elliptic system completes the paper.     

The persistence of ideal shock waves

Local-in-time piecewise smooth solutions to hyperbolic systems of conservation laws are constructed by means of Li-Yu theory. The novelty consists in the application of this approach to shock waves for which the number of outgoing modes is at least as big as the number of incoming modes (undercompressive shocks), the motivation in a possible interpretation from the zero dissipation limit point of view.     

The gap lemma and geometric criteria for instability of viscous shock profiles

An obstacle in the use of Evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis, since the Evans function has in general been constructed only away from the essential spectrum. A notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles. Here we prove a general theorem, the “gap lemma,” concerning the analytic continuation of the Evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave. This allows geometric stability theory to be applied in many cases where it could not be applied previously. We demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves. A necessary geometric condition for stability is determined in terms of the sign of a certain Melnikov integral of the associated viscous profile. This sign can easily be evaluated numerically. We also compute it analytically for solutions of several important classes of systems. In particular, we show for a wide class of systems that homoclinic (solitary) waves are linearly unstable, confirming these as the first known examples of unstable viscous shock waves. We also show that (strong) heteroclinic undercompressive waves are sometimes unstable. Similar stability conditions are also derived for Lax and overcompressive shocks and for n × n conservation laws, n ≥ 2. © 1998 John Wiley & Sons, Inc.     

Stability of undercompressive shock profiles

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed under-overcompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations |u₀(x)|?E₀(1+|x|)^−3/2 in C0+α, E₀ sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile. This completes the program initiated by Zumbrun and Howard in [K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741-871], extending to the general undercompressive case results obtained for Lax and overcompressive shock profiles in [A. Szepessy, Z. Xin, Nonlinear stability of viscous shock waves, Arch. Ration. Mech. Anal. 122 (1993) 53-103; T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws, Comm. Pure Appl. Math. 50 (11) (1997) 1113-1182; K. Zumbrun, P. Howard, Pointwise semigroup methods and stability of viscous shock waves, Indiana Univ. Math. J. 47 (4) (1998) 741-871; K. Zumbrun, Refined wave-tracking and nonlinear stability of viscous Lax shocks, Methods Appl. Anal. 7 (2000) 747-768; M.-R. Raoofi, L¹-asymptotic behavior of perturbed viscous shock profiles, thesis, Indiana Univ., 2004; C. Mascia, K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks, Indiana Univ. Math. J. 51 (4) (2002) 773-904; C. Mascia, K. Zumbrun, Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems, Comm. Pure Appl. Math. 57 (7) (2004) 841-876; C. Mascia, K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity, Arch. Ration. Mech. Anal. 169 (3) (2003) 177-263; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (1) (2004) 93-131; C. Mascia, K. Zumbrun, Stability of large-amplitude shock profiles of general relaxation systems, SIAM J. Math. Anal., in press], and for special “weakly coupled” (respectively scalar diffusive-dispersive) undercompressive profiles in [T.P. Liu, K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation, Comm. Math. Phys. 168 (1) (1995) 163-186; T.P. Liu, K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves, Comm. Math. Phys. 174 (2) (1995) 319-345] (respectively [P. Howard, K. Zumbrun, Pointwise estimates for dispersive-diffusive shock waves, Arch. Ration. Mech. Anal. 155 (2000) 85-169]). In particular, together with spectral results of [K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity-capillarity, SIAM J. Appl. Math. 60 (2000) 1913-1924], our results yield nonlinear stability of large-amplitude undercompressive phase-transitional profiles near equilibrium of Slemrod's model [M. Slemrod, Admissibility criteria for propagating phase boundaries in a van der Waals fluid, Arch. Ration. Mech. Anal. 81 (4) (1983) 301-315] for van der Waal gas dynamics or elasticity with viscosity-capillarity.     

EXISTENCE AND STABILITY OF VISCOUS SHOCK PROFILES FOR 2-D ISENTROPIC MHD WITH INFINITE ELECTRICAL RESISTIVITY

For the two-dimensional Navier-Stokes equations of isentropic magnetohydrodynamics(MHD)withγ-law gas equation of state,γ≥1,and infinite electrical resistivity,we carry out a global analysis categorizing all possible viscous shock profiles.Precisely,we show that the phase portrait of the traveling-wave ODE generically consists of either two rest points connected by a viscous Lax profile,or else four rest points,two saddles and two nodes.In the latter configuration,which rest points are connected by profiles depends on the ratio of viscosities,and can involve Lax,overcompressive,or undercompressive shock profiles.Considered as three-dimensional solutions,undercompressive shocks are Lax-type(Alfven)waves.For the monatomic and diatomic casesγ=5/3 andγ=7/5,with standard viscosity ratio for a nonmagnetic gas,we find numerically that the the nodes are connected by a family of overcompressive profiles bounded by Lax profiles connecting saddles to nodes,with no undercompressive shocks occurring.We carry out a systematic numerical Evans function analysis indicating that all of these two-dimensional shock profiles are linearly and nonlinearly stable,both with respect to two-and three-dimensional perturbations.For the same gas constants,but different viscosity ratios,we investigate also cases for which undercompressive shocks appear;these are seen numerically to be stable as well,both with respect to two-dimensional and(in the neutral sense of convergence to nearby Riemann solutions)three-dimensional perturbations.     

One-dimensional cylindrical shock waves in non-ideal gas under magnetic field

In the present paper, we analyze the evolutionary behavior of imploding strong shock waves propagating through a non-ideal gas in the presence of axial magnetic field. An evolution equation has been constructed by using the method based on the kinematics of one-dimensional motion of shock waves. The values of similarity exponents have been calculated by using the first order truncation approximation which describes the decay behavior of strong shocks. The approximate values of the similarity exponents are compared with the similarity exponents calculated by the CCW approximation, the exact similarity solution and perturbation technique.

     

Modulation theory solution for nonlinearly resonant,fifth‐order Korteweg–de Vries,nonclassical, traveling dispersive shock waves

A new class of resonant dispersive shock waves was recently identified as solutions of the Kawahara equation— a Korteweg–de Vries (KdV) type nonlinear wave equation with third‐ and fifth‐order spatial derivatives— in the regime of nonconvex, linear dispersion. Linear resonance resulting from the third‐ and fifth‐order terms in the Kawahara equation was identified as the key ingredient for nonclassical dispersive shock wave solutions. Here, nonlinear wave (Whitham) modulation theory is used to construct approximate nonclassical traveling dispersive shock wave (TDSW) solutions of the fifth‐ order KdV equation without the third derivative term, hence without any linear resonance. A self‐similar, simple wave modulation solution of the fifth order, weakly nonlinear KdV–Whitham equations is obtained that matches a constant to a heteroclinic traveling wave via a partial dispersive shock wave so that the TDSW is interpreted as a nonlinear resonance. The modulation solution is compared with full numerical solutions, exhibiting excellent agreement. The TDSW is shown to be modulationally stable in the presence of sufficiently small third‐order dispersion. The Kawahara–Whitham modulation equations transition from hyperbolic to elliptic type for sufficiently large third‐order dispersion, which provides a possible route for the TDSW to exhibit modulational instability.     

Modulational Instability in the Whitham Equation for Water Waves

We show that periodic traveling waves with sufficiently small amplitudes of the Whitham equation, which incorporates the dispersion relation of surface water waves and the nonlinearity of the shallow water equations, are spectrally unstable to long‐wavelengths perturbations if the wave number is greater than a critical value, bearing out the Benjamin–Feir instability of Stokes waves; they are spectrally stable to square integrable perturbations otherwise. The proof involves a spectral perturbation of the associated linearized operator with respect to the Floquet exponent and the small‐amplitude parameter. We extend the result to related, nonlinear dispersive equations.     

Exact solutions of an integro-differential equation with quadratically cubic nonlinearity

Exact solutions of a nonlinear integro-differential equation with quadratically cubic nonlinear term are found. The equation governs, in particular, stationary shock wave propagation in relaxing media. For the exponential kernel the shapes of both compression and rarefaction shocks having a finite width of the front are calculated. For media with limited “memorizing time” the difference relation permitting the construction of wave profile by the mapping method is derived. The initial equation is rather general. It governs the evolution of nonlinear waves in real distributed systems, for example, in biological tissues, structurally inhomogeneous media and in some meta-materials.     

Some Results on the Stability of Non-classical Shock Waves

Part 1 of this paper establishes the infinite-time stability of a class of over-compressive viscous shock waves; the equations studied here are a mathematical analogue of those of magnetohydrodynamics. Part 2 communicates a rather general short-time stability result for undercompressive shock waves in several space dimensions; technically, this is an easy extension of Majda's corresponding result for Laxian shock waves.     

Quasi-classical approach to the inverse scattering problem for the KdV equation,and solution of the Whitham modulation equations

We consider an initial value problem for the KdV equation in the limit of weak dispersion. This model describes the formation and evolution in time of a nondissipative shock wave in plasma. Using the perturbation theory in power series of a small dispersion parameter, we arrive at the Riemann simple wave equation. Once the simple wave is overturned, we arrive at the system of Whitham modulation equations that describes the evolution of the resulting nondissipative shock wave. The idea of the approach developed in this paper is to study the asymptotic behavior of the exact solution in the limit of weak dispersion, using the solution given by the inverse scattering problem technique. In the study of the problem, we use the WKB approach to the direct scattering problem and use the formulas for the exact multisoliton solution of the inverse scattering problem. By passing to the limit, we obtain a finite set of relations that connects the space-time parameters x, t and the modulation parameters of the nondissipative shock wave.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 44–61, January, 1996.     

A MUSCL smoothed particles hydrodynamics for compressible multi‐material flows

By incorporating the Monotone Upwind Scheme of Conservation Law (MUSCL) scheme into the smoothed particles hydrodynamics (SPH) method and making use of an interparticle contact algorithm, we present a MUSCL–SPH scheme of second order for multifluid computations, which extends the Riemann‐solved‐based SPH method. The numerical tests demonstrate high accuracy and resolution of the scheme for both shocks, contact discontinuities, and rarefaction waves in the one‐dimensional shock tube problem. For the two‐dimensional cylindrical Noh and shock‐bubble interaction problems, the MUSCL–SPH scheme can resolve shocks well. Copyright © 2015 John Wiley & Sons, Ltd.     

The ultra‐relativistic Euler equations

We study the ultra‐relativistic Euler equations for an ideal gas, which is a system of nonlinear hyperbolic conservation laws. We first analyze the single shocks and rarefaction waves and solve the Riemann problem in a constructive way. Especially, we develop an own parametrization for single shocks, which will be used to derive a new explicit shock interaction formula. This shock interaction formula plays an important role in the study of the ultra‐relativistic Euler equations. One application will be presented in this paper, namely, the construction of explicit solutions including shock fronts, which gives an interesting example for the non‐backward uniqueness of our hyperbolic system. Copyright © 2014 John Wiley & Sons, Ltd.     

Turbulent shock waves in a collisionfree plasma

The paper deals with the structure of collisionless shocks arising from turbulent wave-particle interactions. The conditions under which wave-particle interaction effects could become significant leading to growing waves and a shock are discussed. Using the Mott-Smith expression for the zero-order distribution functions for the ions within the shock, the dielectric constant as well as the integral representing the wave-particle interaction term in the Lenard-Balescu equation are evaluated for a collisionless plasma. An expression is given for the ion distribution function within the shock. It is shown that the component of the pressure tensor perpendicular to the direction of flow of the plasma leads to a new kind of viscosity term arising from the interaction of the particles with the growing waves and this provides a dissipative mechanism to account for the conversion of the kinetic energy of the incoming plasma into the thermal energy of the hot ionised gas behind the shock.     

A fully discrete scheme for diffusive-dispersive conservation laws

Summary.   We introduce a fully discrete (in both space and time) scheme for the numerical approximation of diffusive-dispersive hyperbolic conservation laws in one-space dimension. This scheme extends an approach by LeFloch and Rohde [4]: it satisfies a cell entropy inequality and, as a consequence, the space integral of the entropy is a decreasing function of time. This is an important stability property, shared by the continuous model as well. Following Hayes and LeFloch [2], we show that the limiting solutions generated by the scheme need not coincide with the classical Oleinik-Kruzkov entropy solutions, but contain nonclassical undercompressive shock waves. Investigating the properties of the scheme, we stress various similarities and differences between the continuous model and the discrete scheme (dynamics of nonclassical shocks, nucleation, etc). Received November 15, 1999 / Revised version received May 27, 2000 / Published online March 20, 2001     

DIFFUSIVE—DISPERSIVE TRAVELING WAVES AND KINETIC RELATIONS IV. COMPRESSIBLE EULER EQUATIONS

The authors consider the Euler equations for a compressible fluid in one space dimension when the equation of state of the fluid does not fulfill standard convexity assumptions and viscosity and capillarity effects are taken into account. A typical example of nonconvex constitutive equation for fluids is Van der Waals' equation. The first order terms of these partial differential equations form a nonlinear system of mixed (hyperbolic-elliptic) type. For a class of nonconvex equations of state, an existence theorem of traveling waves solutions with arbitrary large amplitude is established here. The authors distinguish between classical (compressive) and nonclassical (undercompressive) traveling waves. The latter do not fulfill Lax shock inequalities, and are characterized by the so-called kinetic relation, whose properties are investigated in this paper.