Symmetric Hyperbolic-parabolic Systems Research Articles

Hyperbolic–parabolic normal form and local classical solutions for cross-diffusion systems with incomplete diffusion

We investigate degenerate cross-diffusion equations, with a rank-deficient diffusion-matrix, modelling multispecies population dynamics driven by partial pressure gradients. These equations have recently been found to arise in a mean-field limit of interacting stochastic particle systems. To date, their analysis in multiple space dimensions has been confined to the purely convective case with equal mobility coefficients. In this article, we introduce a normal form for an entropic class of such equations which reveals their structure of a symmetric hyperbolic–parabolic system. Due to the state-dependence of the range and kernel of the singular diffusive matrix, our way of rewriting the equations is different from that classically used for symmetric second-order systems with a nullspace invariance property. By means of this change of variables, we solve the Cauchy problem for short times and positive initial data in for

Global well-posedness and time-decay of solutions for the compressible Hall-magnetohydrodynamic system in the critical Besov framework

We consider the global existence of solution for the initial value problem for the compressible Hall-magnetohydrodynamic system in the whole space R3. The system consists of a hyperbolic-parabolic system of partial differential equations of the conservation laws type with non-symmetric diffusion. We show the existence of solution as a perturbation from a constant equilibrium state (ρ¯,0,B¯), where ρ¯>0 is a constant density, 0∈R3 is the zero velocity and B¯∈R3 is a constant magnetic field. The time-decay of the solution in the Besov spaces is also established. Our results show the pointwise estimate of the solution in the Fourier space for the linearized Hall-MHD system that related to the result obtained by Umeda–Kawashima–Shizuta [33] for a general class of linear symmetric hyperbolic-parabolic systems with symmetric diffusion. We utilize a systematic use of the product estimates in the Chemin–Lerner spaces and apply the energy method due to Matsumura–Nishida [27].

Uniform dissipativity for mixed-order hyperbolic systems, with an application to relativistic fluid dynamics

This paper extends a characterization of uniform dissipativity given in a previous publication by two of the authors for second-order hyperbolic systems to an equally natural class of hyperbolic systems of mixed order. These systems are finite-speed-of-propagation counterparts of symmetric hyperbolic-parabolic systems, and the characterizing condition obtained plays a role analogous to the Kawashima condition, inducing decay rates for all Fourier modes in a way that gives – at least for linear constant-coefficients problems, to which this paper restricts attention – uniform decay of solutions in L2 based Sobolev spaces.

Dissipative structure for symmetric hyperbolic-parabolic systems with Korteweg-type dispersion

In this paper, we are concerned with generally symmetric hyperbolic-parabolic systems with Korteweg-type dispersion. Referring to those classical efforts by Kawashima et al., we formulate new structural conditions for the Korteweg-type dispersion and develop the dissipative mechanism of “regularity-gain type.” As an application, it is checked that several concrete model systems (e.g., the compressible Navier-Stokes(-Fourier)-Korteweg system) satisfy the general structural conditions. In addition, the optimality of our general theory on the dissipative structure is also verified by calculating the asymptotic expansions of eigenvalues.

Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.

Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space

In the present paper, we study a system of viscous conservation laws, which is rewritten to a symmetric hyperbolic-parabolic system, in one-dimensional half space. For this system, we derive a convergence rate of the solutions towards the corresponding stationary solution with/without the stability condition. The essential ingredient in the proof is to obtain the a priori estimate in the weighted Sobolev space. In the case that all characteristic speeds are negative, we show the solution converges to the stationary solution exponentially if an initial perturbation belongs to the exponential weighted Sobolev space. The algebraic convergence is also obtained in the similar way. In the case that one characteristic speed is zero and the other characteristic speeds are negative, we show the algebraic convergence of solution provided that the initial perturbation belongs to the algebraic weighted Sobolev space. The Hardy type inequality with the best possible constant plays an essential role in deriving the optimal upper bound of the convergence rate. Since these results hold without the stability condition, they immediately mean the asymptotic stability of the stationary solution even though the stability condition does not hold.

Existence and asymptotic stability of stationary waves for symmetric hyperbolic–parabolic systems in half-line

In this paper, we consider a one-dimensional half-space problem for a system of viscous conservation laws which is deduced to a symmetric hyperbolic–parabolic system under assuming that the system has a strictly convex entropy function. We firstly prove existence of a stationary solution by assuming that a boundary strength is sufficiently small. The existence of the stationary solution is characterized by the number of negative characteristics. In the case where one characteristic speed is zero at spatial asymptotic state [Formula: see text], we assume that the characteristic field corresponding to the characteristic speed 0 is genuinely nonlinear in order to show existence of a degenerate stationary solution with the aid of a center manifold theory. We next prove that the stationary solution is time asymptotically stable under a smallness assumption on an initial perturbation in the Sobolev space. The key to proof is to derive the uniform a priori estimates by using the energy method, where the stability condition of Shizuta–Kawashima type plays an essential role.

Global smooth flows for compressible Navier–Stokes–Maxwell equations

Umeda et al. (Jpn J Appl Math 1:435–457, 1984) considered a rather general class of symmetric hyperbolic–parabolic systems: $$A^{0}z_{t}+\sum_{j=1}^{n}A^{j}z_{x_{j}}+Lz=\sum_{j,k=1}^{n}B^{jk}z_{x_{j}x_{k}}$$ and showed optimal decay rates with certain dissipative assumptions. In their results, the dissipation matrices $${L}$$ and $${B^{jk}(j,k=1,\ldots,n)}$$ are both assumed to be real symmetric. So far there are no general results in case that $${L}$$ and $${B^{jk}}$$ are not necessarily symmetric, which is left open now. In this paper, we investigate compressible Navier–Stokes–Maxwell (N–S–M) equations arising in plasmas physics, which is a concrete example of hyperbolic–parabolic composite systems with non-symmetric dissipation. It is observed that the Cauchy problem for N–S–M equations admits the dissipative mechanism of regularity-loss type. Consequently, extra higher regularity is usually needed to obtain the optimal decay rate of $${L^{1}({\mathbb{R}}^3)}$$ - $${L^2({\mathbb{R}}^3)}$$ type, in comparison with that for the global-in-time existence of smooth solutions. In this paper, we obtain the minimal decay regularity of global smooth solutions to N–S–M equations, with aid of $${L^p({\mathbb{R}}^n)}$$ - $${L^{q}({\mathbb{R}}^n)}$$ - $${L^{r}({\mathbb{R}}^n)}$$ estimates. It is worth noting that the relation between decay derivative orders and the regularity index of initial data is firstly found in the optimal decay estimates.

Energy estimate for a linear symmetric hyperbolic-parabolic system in half line

In the present paper, we study the initial boundary value problem for a linear symmetric hyperbolic-parabolic system in one-dimensional half space. We obtain a priori estimates by using an energy method developed by Matsumura--Nishida for half space problem under the assumption that a stability condition of Shizuta--Kawashima type holds. The method developed in the present paper is applicable to showing the nonlinear stability of boundary layer solutions for a system of viscous conservation laws in half space.

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 | u 0 ( x ) | ⩽ E 0 ( 1 + | x | ) − 3 / 2 in C 0 + α , E 0 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, L 1 -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.

The local Cauchy problem for ionized magnetized reactive gas mixtures

We investigate a system of partial differential equations modelling ionized magnetized reactive gas mixtures. In this model, dissipative fluxes are anisotropic linear combinations of fluid variable gradients and also include zeroth-order contributions modelling the direct effect of electromagnetic forces. There are also gradient dependent source terms like the conduction current in the Maxwell–Ampere equation. We introduce the notion of partial symmetrizability and that of entropy for such systems of partial differential equations and establish their equivalence. By using entropic variables, we recast the system into a partially normal form, that is, in the form of a quasilinear partially symmetric hyperbolic–parabolic system. Using a result of Vol'Pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution. Copyright © 2005 John Wiley & Sons, Ltd.

Le problème de Cauchy local pour les plasmas dissipatifs

We investigate a system of partial differential equations modeling dissipative plasmas. Transport fluxes are anisotropic linear combinations of gradients and also include zeroth order contributions due to electromagnetic forces. There are also source terms depending on the solution gradient. By using entropic variables, we first recast the system in a partially symmetric form and next in the form of a quasilinear partially symmetric hyperbolic-parabolic system. Using a result of Vol'Pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem. To cite this article: V. Giovangigli, B. Graille, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Stability of small‐amplitude shock profiles of symmetric hyperbolic‐parabolic systems

AbstractCombining pointwise Green's function bounds obtained in a companion paper [36] with earlier, spectral stability results obtained in [16], we establish nonlinear orbital stability of small‐amplitude Lax‐type viscous shock profiles for the class of dissipative symmetric hyperbolic‐parabolic systems identified by Kawashima [20], notably including compressible Navier‐Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in Lp with respect to small L1 ∩ H3 perturbations, 2 ≤ p ≤ ∞. Our analysis extends and somewhat refines the approach introduced in [35] to treat stability of relaxation profiles. © 2004 Wiley Periodicals, Inc.

Stability of Large-Amplitude Viscous Shock Profiles of Hyperbolic-Parabolic Systems

We establish nonlinear L1∩H3→Lp orbital stability, 2≦p≤∞, with sharp rates of decay, of large-amplitude Lax-type shock profiles for a class of symmetric hyperbolic-parabolic systems including compressible gas dynamics and magnetohydrodynamics (MHD) under the necessary conditions of strong spectral stability, i.e., a stable point spectrum of the linearized operator about the wave, transversality of the profile, and hyperbolic stability of the associated ideal shock. This yields in particular, together with the spectral stability results of [50], the nonlinear stability of arbitrarily large-amplitude shock profiles of isentropic Navier–Stokes equations for a gamma-law gas as γ→1: the first complete large-amplitude stability result for a shock profile of a system with real (i.e., partial) viscosity. A corresponding small-amplitude result was established in [53, 54] for general systems of Kawashima class by a combination of ‘‘Kawashima-type’’ energy estimates and pointwise Green function bounds, where the small-amplitude assumption was used only to close the energy estimates. Here, under the mild additional assumption that hyperbolic characteristic speeds (relative to the shock) are not only nonzero but of a common sign, we close the estimates instead by use of a Goodman-type weighted norm [25, 26] designed to control estimates in the crucial hyperbolic modes.

Spectral stability of small-amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems

We prove one-dimensional spectral stability of small amplitude shock profiles for degenerate viscosity conservation laws that are dissipative symmetric hyperbolic-parabolic in the sense of Kawashima [Ka.1], generalizing well-known results of Matsumura and Nishihara [MN] and of Kawashima, Matsumura, and Nishihara [KM,KMN] for the case of compressible gas dynamics. The proof follows an approach different from that of [MN,KM,KMN], instead combining the weighted energy method of Goodman [Go.2] with the auxiliary derivative estimates of Kawashima [Ka.1].

The local Cauchy problem for multicomponent reactive flows in full vibrational non-equilibrium

We consider reactive mixtures of dilute polyatomic gases in full vibrational non-equilibrium. The governing equations are derived from the kinetic theory and possesses an entropy. We recast this system of conservation laws into a symmetric conservative form by using entropic variables. Following a formalism developed by the authors in a previous paper, the system is then rewritten into a normal form, that is, in the form of a quasilinear symmetric hyperbolic-parabolic system. Using a result of Vol'pert and Hudjaev, we prove local existence and uniqueness of a bounded smooth solution to the Cauchy problem.

On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws

On considere le probleme aux valeurs initiales pour les systemes non lineaires d'ordre 2: w t +∑ j=1 n f j (w) x j =∑ i,j=1 n {G ij (w) w xj } x i ou G ij (w) sont des matrices m×m. On donne une condition suffisante pour que ce systeme se mette sous une forme normale d'un systeme hyperbolique-parabolique

Smooth global solutions for two-dimensional equations of electro-magneto-fluid dynamics

The equations of an electrically conducting compressible fluid in electro-magneto-fluid dynamics are studied. It is proved that in a certain case of two-dimensional flow, the equations of the fluid become a symmetric hyperbolic-parabolic system in both of the viscous and non-viscous cases. Therefore, the initial value problem is well posed in the Sobolev spaces at least for short time interval. Furthermore, in the viscous case, the solution exists globally in time and tends to the constant state as time goes to infinity, provided the initial data are closed to the constant state. The proof is based on a technical energy method, which makes use of a quadratic function associated with the total energy of the fluid.