Uniform Stability Estimates for Constant-Coefficient Symmetric Hyperbolic Boundary Value Problems

Answering a question left open in Métivier and Zumbrun (2005 Métivier , G. , Zumbrun , K. ( 2005 ). Variable multiplicities, hyperbolic boundary value problems for symmetric systems with variable multiplicities . J. Diff. Eq. 211 ( 1 ): 61 – 134 . [Google Scholar]), we show for general symmetric hyperbolic boundary problems with constant coefficients, including in particular systems with characteristics of variable multiplicity, that the uniform Lopatinski condition implies strong L ² well-posedness, with no further structural assumptions. The result applies, more generally, to any system that is strongly L ² well-posed for at least one boundary condition. The proof is completely elementary, avoiding reference to Kreiss symmetrizers or other specific techniques. On the other hand, it is specific to the constant-coefficient case; at least, it does not translate in an obvious way to the variable-coefficient case. The result in the hyperbolic case is derived from a more general principle that can be applied, for example, to parabolic or partially parabolic problems like the Navier–Stokes or viscous MHD equations linearized about a constant state or even a viscous shock.     

Initial Boundary Value Problems for Quasilinear Hyperbolio- Parabolio Ooupled Systems in Higher Dimensional Spaces

The initial bounary value problem for quasilinear byperbolie-parabolic coupled systems in higher dimensional spaces, which arises in many mechanical problerns is considered. Under the assumptions that the-hyperbolic part of the coupled system is a quasilinear symmetric hyperbolic system and the parabolic part is a quasilinear parabolic system of second order and suitable assumptions of smoothness and compatibiliy conditions, the existence and uniqueness of local smooth solution is proved in the cases that the boundary of domain is noncharacteristic or uniformly characteristic with respect to the hyperbolic part. As an application, the existence and uniqueness of local smooth solution for the initial boundary problem of the radiation hydrodynamic system, as well as of the viscous compressible hydrodynamic system, with solid wal1 boundary, is obtained.     

A coupled nonlinear hyperbolic-parabolic system

A boundary initial value problem for a quasi-linear hyperbolic system in one space variable is coupled to a boundary initial value problem for a quasi-linear parabolic equation in two space variables. The coupling occurs through one of the boundary conditions for the hyperbolic system and the source term in the parabolic equation. Such a coupling can arise in the consideration of gas flowing in a porous medium and out of that medium via a pipe. A local existence and uniqueness theorem is demonstrated. The proof involves the method of characteristics, Bernstein's estimates for parabolic partial differential equations, and the contracting mapping theorem.     

Hyperbolic boundary value problems for symmetric systems with variable multiplicities

We extend the Kreiss-Majda theory of stability of hyperbolic initial-boundary-value and shock problems to a class of systems, notably including the equations of magnetohydrodynamics (MHD), for which Majda's block structure condition does not hold: namely, simultaneously symmetrizable systems with characteristics of variable multiplicity, satisfying at points of variable multiplicity either a “totally nonglancing” or a “nonglancing and linearly splitting” condition. At the same time, we give a simple characterization of the block structure condition as “geometric regularity” of characteristics, defined as analyticity of associated eigenprojections. The totally nonglancing or nonglancing and linearly splitting conditions are generically satisfied in the simplest case of crossings of two characteristics, and likewise for our main physical examples of MHD or Maxwell equations for a crystal. Together with previous analyses of spectral stability carried out by Gardner-Kruskal and Blokhin-Trakhinin, this yields immediately a number of new results of nonlinear inviscid stability of shock waves in MHD in the cases of parallel or transverse magnetic field, and recovers the sole previous nonlinear result, obtained by Blokhin-Trakhinin by direct “dissipative integral” methods, of stability in the zero-magnetic field limit. We also discuss extensions to the viscous case.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

This is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Unique solvability of a non‐local problem for mixed‐type equation with fractional derivative

In this work, we investigate a boundary problem with non‐local conditions for mixed parabolic–hyperbolic‐type equation with three lines of type changing with Caputo fractional derivative in the parabolic part. We equivalently reduce considered problem to the system of second kind Volterra integral equations. In the parabolic part, we use solution of the first boundary problem with appropriate Green's function, and in hyperbolic parts, we use corresponding solutions of the Cauchy problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

In this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Asymptotic expansions for hyperbolic–parabolic systems with a small parameter

In this paper we consider initial-boundary value problems for systems with a small parameter ?. The problems are mixed hyperbolic–parabolic when ? > 0 and hyperbolic when ? = 0. Often the solution can be expanded asymptotically in ? and to first approximation it consists of the solution of the corresponding hyperbolic problem and a boundary layer part. We prove sufficient conditions for the expansion to exist and give estimates of the remainder. We also examine how the boundary conditions should be choosen to avoid O(1) boundary layers.     

Thermoelasticity with second sound—exponential stability in linear and non‐linear 1‐d

We consider linear and non‐linear thermoelastic systems in one space dimension where thermal disturbances are modelled propagating as wave‐like pulses travelling at finite speed. This removal of the physical paradox of infinite propagation speed in the classical theory of thermoelasticity within Fourier's law is achieved using Cattaneo's law for heat conduction. For different boundary conditions, in particular for those arising in pulsed laser heating of solids, the exponential stability of the now purely, but slightly damped, hyperbolic linear system is proved. A comparison with classical hyperbolic–parabolic thermoelasticity is given. For Dirichlet type boundary conditions—rigidly clamped, constant temperature—the global existence of small, smooth solutions and the exponential stability are proved for a non‐linear system. Copyright © 2002 John Wiley & Sons, Ltd.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

时滞抛物型与双曲型微分方程解的振动性的讨论

对一类线性以及非线性抛物型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,给出了解振动的一些结论.并且对一类线性以及带强迫项的非线性双曲型时滞微分方程的解在第一或第二边值条件下解振动的充分必要条件进行了讨论,也给出了一些结论.     

On the well-posedness of the cauchy problem and the mixed problem for some class of hyperbolic systems and equations with constant coefficients and characteristics of variable multiplicity

We study the well-posedness of the mixed problem for hyperbolic equations with constant coefficients and with characteristics of variable multiplicity. We single out a class of higher-order hyperbolic operators with constant coefficients and with characteristics of variable multiplicity, for which we obtain a generalization of the Sakamoto conditions for the well-posedness of the mixed problem in L ₂.     

Viscous stress relaxation,characteristics, and numerical boundary conditions

The anomalous infinite propagation speeds in the classical parabolic flow equations are removed by the inclusion of a small amount of fluid elasticity or viscous stress relaxation. The inclusion of such effects results in a hyperbolic system of equations with a complete set of characteristic equations. The directional characteristic equations are used to give insights into the appropriate boundary molecules to be used in finite difference numerical schemes. © 1994 John Wiley & Sons, Inc.     

Resonantly Interacting Weakly Nonlinear Hyperbolic Waves. I. A Single Space Variable

We present a systematic asymptotic theory for resonantly interacting weakly nonlinear hyperbolic waves in a single space variable. This theory includes as a special case 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. 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. (In the general case we give many new conditions guaranteeing nonresonance for a given hyperbolic system with prescribed initial data, as well as other new structural conditions which imply that resonance occurs.) A method for treating resonantly interacting waves in several space variables, together with applications, will be developed by the authors elsewhere.     

Well-posedness of the Hydrodynamic Model for Semiconductors

This paper concerns the well-posedness of the hydrodynamic model for semiconductor devices, a quasi-linear elliptic–parbolic–hyperbolic system. Boundary conditions for elliptic and parabolic equations are Dirichlet conditions while boundary conditions for the hyperbolic equations are assumed to be well-posed in L² sense. Maximally strictly dissipative boundary conditions for the hyperbolic equations satisfy the assumption of well-posedness in L² sense. The well-posedness of the model under the boundary conditions is demonstrated.     

DOMAIN DECOMPOSITION PRECONDITIONERS FOR SECOND-ORDER HYPERBOLIC EQUATIONS ON L-SHAPED REGIONS

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations on L-shaped region are considered. We analyse the use of domain deocmposilion preconditioner.s for the solution of linear systems via the preconditioned conjugate gradient method. For the constant-coefficient second-order hyperbolic equaions with initial and Dirichlet boundary conditions,we prove that the conditionnumber of the preconditioned interface system is bounded by 2+x2 2+0.46x2 where x is the quo-tient between the lime and space steps. Such condition number produces a convergence rale that is independent of gridsize and aspect ratios. The results could be extended to parabolic equations.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232     

A Taylor expansion approach for solving partial differential equations with random Neumann boundary conditions

Nonlinear partial differential equation with random Neumann boundary conditions are considered. A stochastic Taylor expansion method is derived to simulate these stochastic systems numerically. As examples, a nonlinear parabolic equation (the real Ginzburg-Landau equation) and a nonlinear hyperbolic equation (the sine-Gordon equation) with random Neumann boundary conditions are solved numerically using a stochastic Taylor expansion method. The impact of boundary noise on the system evolution is also discussed.     

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.