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. Our methods apply also to the viscous case.
In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in \cite{Garetto2018}. In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.
In this paper we study first order hyperbolic systems with multiple characteristics (weakly hyperbolic) and time-dependent analytic coefficients. The main question is when the Cauchy problem for such systems is well-posed in $C^{\infty}$ and in $\mathcal{D}'$. We prove that the analyticity of the coefficients combined with suitable hypotheses on the eigenvalues guarantee the $C^\infty$ well-posedness of the corresponding Cauchy problem. This result is an extension to systems of the analogous results for scalar equations recently obtained by Jannelli and Taglialatela in \cite{JT} and by the aut…
We study hyperbolic systems with multiplicities and smo\-oth coefficients. In the case of non-analytic, smooth coefficients, we prove well-posedness in any Gevrey class and when the coefficients are analytic, we prove $C^\infty$ well-posedness. The proof is based on a transformation to block Sylvester form introduced by D'Ancona and Spagnolo in Ref. 9 which increases the system size but does not change the eigenvalues. This reduction introduces lower order terms for which appropriate Levi-type conditions are found. These translate then into conditions on the original coefficient matrix. This p…
Extending investigations of Métivier&Zumbrun in the hyperbolic case, we treat stability of viscous shock and boundary layers for viscous perturbations of multidimensional hyperbolic systems with characteristics of variable multiplicity, specifically the construction of symmetrizers in the low-frequency regime where variable multiplicity plays a role. At the same time, we extend the boundary-layer theory to ``real'' or partially parabolic viscosities, Neumann or mixed-type parabolic boundary conditions, and systems with nonconservative form, in addition proving a more fundamental version of the…
This paper contributes to the wider study of hyperbolic equations with multiplicities. We focus here on some classes of higher order hyperbolic equations with space dependent coefficients in any space dimension. We prove Sobolev well-posedness of the corresponding Cauchy problem (with loss of derivatives due to the multiplicities) under suitable Levi conditions on the lower order terms. These conditions generalise the well known Olienik's conditions in \cite{O70} to orders higher than $2$.
In this paper we analyse the well-posedness of the Cauchy problem for a rather general class of hyperbolic systems with space-time dependent coefficients and with multiple characteristics of variable multiplicity. First, we establish a well-posedness result in anisotropic Sobolev spaces for systems with upper triangular principal part under interesting natural conditions on the orders of lower order terms below the diagonal. Namely, the terms below the diagonal at a distance $k$ to it must be of order $-k$. This setting also allows for the Jordan block structure in the system. Second, we give …
We establish long-time stability of multi-dimensional viscous shocks of a general class of symmetric hyperbolic--parabolic systems with variable multiplicities, notably including the equations of compressible magnetohydrodynamics (MHD) in dimensions $d\ge 2$. This extends the existing result established by K. Zumbrun for systems with characteristics of constant multiplicity to the ones with variable multiplicity, yielding the first such a stability result for (fast) MHD shocks. At the same time, we are able to drop a technical assumption on structure of the so--called glancing set that was nec…
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral operators. It is shown that this series is an asymptotic expansion with respect to smoothness under quite general geometric conditions. Propagation of singularities and sharp regularity properties of solutions are obtained. Results are applied to establish regularity estimates for scalar weakly hyperbolic equations with involutive characteristics. They are also ap…
We show the continuity of the flow map for quasilinear symmetric hyperbolic systems with general right--hand sides in different functional setting, including weighted Sobolev spaces $H_{s,δ}$. An essential tool to achieve the continuity of the flow map is a low regularity energy estimate. We then apply these results to the Euler--Poisson system which describes various systems of physical interests.
We study existence, uniqueness, and distributional aspects of generalized solutions to the Cauchy problem for first-order symmetric (or Hermitian) hyperbolic systems of partial differential equations with Colombeau generalized functions as coefficients and data. The proofs of solvability are based on refined energy estimates on lens-shaped regions with spacelike boundaries. We obtain several variants and also partial extensions of previous results and provide aspects accompanying related recent work by C. Garetto and M. Oberguggenberger.
We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems…
This paper is concerned with the decay structure for linear symmetric hyperbolic systems with relaxation. When the relaxation matrix is symmetric, the dissipative structure of the systems is completely characterized by the Kawashima-Shizuta stability condition formulated in \cite{UKS84,SK85}, and we obtain the asymptotic stability result together with the explicit time-decay rate under that stability condition. However, some physical models which satisfy the stability condition have non-symmetric relaxation term (cf.~the Timoshenko system and the Euler-Maxwell system). Moreover, it had been al…
Answering a question left open in \cite{MZ2}, 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 con…
We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an extension of earlier work by the authors for the same problem in one space dimension.
On manifolds with an even Riemannian conformally compact Einstein metric, the resolvent of the Lichnerowicz Laplacian, acting on trace-free, divergence-free, symmetric 2-tensors is shown to have a meromorphic continuation to the complex plane, defining quantum resonances of this Laplacian. For higher rank symmetric tensors, a similar result is proven for (convex cocompact) quotients of hyperbolic space.
We consider a characteristic initial value problem for a class of symmetric hyperbolic systems with initial data given on two smooth null intersecting characteristic surfaces. We prove existence of solutions on a future neighborhood of the initial surfaces. The general result is applied to general semilinear wave equations, as well as the Einstein equations with or without sources, and conformal variations thereof.
This note establishes properties of a model of dissipative fluid dynamics as a symmetric hyperbolic system by which Ruggeri once triggered the development of Rational Extended Thermodynamics. These properties, i.e., (1) convergence of solutions to those of the Navier-Stokes-Fourier equations and (2) nonlinear asymptotic stability of homogeneous reference solutions, also hold in the analogous situation for barotropic fluids.
We study the problem of propagation of analytic regularity for semi-linear symmetric hyperbolic systems. We adopt a global perspective and we prove that if the initial datum extends to a holomorphic function in a strip of radius (=width) ε_0, the same happens for the solution u(t,.) for a certain radius ε(t), as long as the solution exists. Our focus is on precise lower bounds on the spatial radius of analyticity ε(t) as t grows. We also get similar results for the Schroedinger equation with a real-analytic electromagnetic potential.
In this paper, a geometric process to compare solutions of symmetric hyperbolic systems on (possibly different) globally hyperbolic manifolds is realized via a family of intertwining operators. By fixing a suitable parameter, it is shown that the resulting intertwining operator preserves Hermitian forms naturally defined on the space of homogeneous solutions. As an application, we investigate the action of the intertwining operators in the context of algebraic quantum field theory. In particular, we provide a new geometric proof for the existence of the so-called Hadamard states on globally hy…
Motivated by the future stability problem of solutions of the Einstein--Yang--Mills (EYM) system with arbitrary dimension, we aim to (1) construct a tensorial symmetric hyperbolic formulation for the $(n+1)$-dimensional EYM system in the temporal gauge; (2) establish the local well-posedness for the Cauchy problem of EYM equations in the temporal gauge using this tensorial symmetric hyperbolic system. By introducing certain auxiliary variables, we extend essentially the $(n+1)$-dimensional Yang--Mills system to a tensorial symmetric hyperbolic system. On the contrary, this symmetric hyperbolic…
Combining pointwise Green's function bounds obtained in a companion paper [MZ.2] with earlier, spectral stability results obtained in [HuZ], we establish nonlinear orbital stability of small amplitude viscous shock profiles for the class of dissipative symmetric hyperbolic-parabolic systems identified by Kawashima [Kaw], notably including compressible Navier--Stokes equations and the equations of magnetohydrodynamics, obtaining sharp rates of decay in $L^p$ with respect to small $L^1\cap H^3$ perturbations, $2\le p\le \infty$. Our analysis follows the approach introduced in [MZ.1] to treat sta…
