GLOBAL EXISTENCE, UNIQUENESS, AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

A global existence theorem is established for an initial-boundary value problem,with time-dependent boundary data,arising in a lumped parameter model of pulse combustion; the model in question gives ri...     

On a hyperbolic-parabolic system arising in magnetoelasticity

The evolution of a magnetoelastic material is described by a nonlinear hyperbolic-parabolic system. We introduce a simplified but nontrivial model and prove the existence of a unique solution to the corresponding initial boundary value problem.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

一个耦合双曲-抛物系统的全局光滑解

考虑一个源自生物学的耦合双曲-抛物模型的初边值问题.当动能函数为非线性函数以及初始值具有小的L~2能量但其H~2能量可能任意大时,得到了初边值问题光滑解的全局存在性和指数稳定性.而且,如果假定非线性动能函数满足一定的条件,在对初值没任何小条件假定下得到光滑解的全局存在性.通过构造一个新的非负凸熵和做精细的能量估计得到了结果的证明.     

流体运动稳定性方程组椭圆特性的分析与应用

利用抛物化稳定方程(PSE)特征分析得知,原始扰动量的线性和非线性PSE整体来说为抛物型.利用PSE的次特征分析证明,对速度U,在亚音速和跨音速区,线性PSE分别为椭圆型和双曲-抛物型;对速度U+u,在亚音速和跨音速区,非线性PSE分别为椭圆型和双曲-抛物型(其中,U和u分别为主流方向的扰动和未扰流速度分量).结论表明,流体运动稳定性方程组的"抛物化"简化,仅把信息的对流扩散传播抛物化,而保留了信息的对流扰动传播特性,PSE实质上是扩散抛物化稳定性方程组.根据特征次特征理论提出了消除PSE剩余椭圆特性的方法,所得结论对线性PSE已有结论一致,并给出了Mach数的影响.同时,进一步给出了消除非线性PSE的剩余椭圆特性的方法.     

一个模拟趋化现象的广义双曲-抛物系统的光滑解的全局分析

考虑一个模拟趋化现象的广义双曲-抛物系统的Cauchy问题,当动能函数为非线性函数且初始值具有小的L~2能量但其H~2能量可能任意大时,得到了全局光滑解的存在性和渐近行为.这些结果推广了以前的关于动能函数为线性函数或初始值具有小的H~2能量情形下的相关结果,首次获得了关于全局光滑大解方面的结果.这些结果的证明基于构造一个新的非负凸熵和做精细的能量估计.     

Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems

Extending to systems of hyperbolic-parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in Lp, 1?p?∞.     

Viscous splitting approximation of mixed hyperbolic-parabolic convection-diffusion equations

Summary. We first analyse a semi-discrete operator splitting method for nonlinear, possibly strongly degenerate, convection-diffusion equations. Due to strong degeneracy, solutions can be discontinuous and are in general not uniquely determined by their data. Hence weak solutions satisfying an entropy condition are sought. We then propose and analyse a fully discrete splitting method which employs a front tracking scheme for the convection step and a finite difference scheme for the diffusion step. Numerical examples are presented which demonstrate that our method can be used to compute physically correct solutions to mixed hyperbolic-parabolic convection-diffusion equations. Received November 4, 1997 / Revised version received June 22, 1998     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

Uniqueness of the solution of the Tricomi-type problem for multidimensional hyperbolic-parabolic equation

A mixed domain is described in which the solution of a Tricomi-type homogeneous problem for a multidimensional hyperbolic-parabolic equation is trivial.     

Maxwell's equations when the charge conservation is not satisfied

Maxwell's equations are overdetermined when the charge conservation equation is not verified. In order to overcome this problem, different methods have been introduced. We notice that they fit into a framework in which a new formulation which we introduce also fits. These formulations can be classified according to the type of the resulting PDE-system as hyperbolic-elliptic, hyperbolic-parabolic and purely hyperbolic. We show that the resolution of Maxwell's equation through the potentials is always equivalent to the purely hyperbolic formulation and that the hyperbolic-parabolic and hyperbolic-elliptic formulations converge to the purely hyperbolic formulation when introducing a parameter which goes to 0.     

The semistrong limit of multipulse interaction in a thermally driven optical system

We consider the semistrong limit of pulse interaction in a thermally driven, parametrically forced, nonlinear Schrödinger (TDNLS) system modeling pulse interaction in an optical cavity. The TDNLS couples a parabolic equation to a hyperbolic system, and in the semistrong scaling we construct pulse solutions which experience both short-range, tail-tail interactions and long-range thermal coupling. We extend the renormalization group (RG) methods used to derive semistrong interaction laws in reaction-diffusion systems to the hyperbolic-parabolic setting of the TDNLS system. A key step is to capture the singularly perturbed structure of the semigroup through the control of the commutator of the resolvent and a re-scaling operator. The RG approach reduces the pulse dynamics to a closed system of ordinary differential equations for the pulse locations.     

The nonlinear dynamics of a cable of controlled length in a stream

In this paper we study transient processes in one-dimensional deformable cables when there is a nonsteady change in length during towing in a stream of liquid or gas. A new hyperbolic-parabolic model is constructed that takes account of the longitudinal waves in the cable and nonlinear effects caused by the change in the length of the cable according to a prescribed law of control of its length. As an illustration we give an analysis of the evolutionary characteristics of a radial winch and the relative velocity of streamline flow around it for two tachograms of a winch simulating realistic working conditions in the operation of towed oceanographic systems. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 93–98.     

An extension ofA-stability to alternating direction implicit methods

Completely implicit, noniterative, finite-difference schemes have recently been developed by several authors for nonlinear, multidimensional systems of hyperbolic and mixed hyperbolic-parabolic partial differential equations. The method of Douglas and Gunn or the method of approximate factorization can be used to reduce the computational problem to a sequence of one-dimensional or alternating direction implicit (ADI) steps. Since the eigenvalues of partial differential equations (for example, the equations of compressible fluid dynamics) are often widely distributed with large imaginary parts,A-stable integration formulas provide ideal time-differencing approximations. In this paper it is shown that if anA-stable linear multistep method is used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation, then one can always construct an ADI scheme by the method of approximate factorization which is alsoA-stable, i.e., unconditionally stable. A more restrictive result is given for three spatial dimensions. Since necessary and sufficient conditions forA-stability can easily be determined by using the theory of positive real functions, the stability analysis of the factored partial difference equations is reduced to a simple algebraic test.The main results of this paper were presented at the SIAM National Meeting, Madison, Wis., May 24 to 26, 1978, and section 9 was part of a presentation at the 751st Meeting of the American Mathematical Society, San Luis Obispo, California, Nov. 11 to 12, 1977.     

Correctness of the Dirichlet problem for a class of multidimensional hyperbolic-parabolic equations

It is shown that the Dirichlet problem in a cylindrical domain for a class of multidimensional hyperbolic-parabolic equations is uniquely solvable. The criterion of uniqueness of a regular solution is obtained.     

On some second-order non-linear systems of a hyperbolic-parabolic type

We consider local solutions to the Cauchy problem for a class of non-linear hyperbolic-parabolic systems generalizing the systems of elasticity and thermoelasticity. Our main purpose is to relax the usual regularity requirements to include the nonclassical solutions into considerations.     

INITIAL-BOUNDARY VALUE PROBLEMS WITH INTERFACE FOR QUASILINEAR HYPERBOLIC-PARABOLIC COUPLED SYSTEMS

The local time existence and uniqueness theorem is proved for mixed initial-boundaryvalue problems with interface for quasilinear hyperbolic-parabolic coupled systems in twoindependent variables.     

Nonuniqueness of the solution of the Tricomi problem for a multidimensional hyperbolic-parabolic equation

We construct examples of multidimensional hyperbolic-parabolic equations for which the homogeneous Tricomi problem has infinitely many nontrivial solutions.     

On some boundary-value problems with a shift on characteristics for a mixed equation of hyperbolic-parabolic type

We prove theorems on the existence and uniqueness of solutions of nonlocal boundary-value problems with shift for mixed second- and third-order equations of hyperbolic-parabolic type.     

Characteristic Boundary Layers for Mixed Hyperbolic-Parabolic Systems in One Space Dimension and Applications to the Navier-Stokes and MHD Equations

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so-called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second-order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems. © 2020 Wiley Periodicals, Inc.