Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

Der vorliegende Band enthált sechs Beitrage, in denen punktuell und exemplarisch der Stand der Forschung beleuchtet wird. Ein Beitrag von Codling und Frasinski beschreibt in erster Linie Experimente zum noch wenig verstandenen Phânomen der dissoziativen Ionisation von Molekülen, bei dem diese in einem intensiven Laserfeld mehrere Elektronén verlieren und in Ionen mit hoher kinetischer Energie zerfallen. Ein wesentlicher Teil dieses Beitrags widmet sich einer von den Autoren entwickelten Auswertungstechnik, welche die Korrelationen zwischen den Fiugzeiten emittierter Ionen und Elektronén sichtbar macht. Es folgen drei Beitrage zur Théorie. Schmelcher und Cederbaum beschreiben Effekte starker statischer Felder, die nur dann zu verstehen sind, wenn die Nicht-Separation der Schwerpunktsbewegung im auBeren Magnetfeld korrekt beriicksichtigt wird. Dies fiihrt

Extended Irreversible Thermodynamics

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

https://iopscience.iop.org/article/10.1088/0034-4885/62/7/201/pdf

Extended irreversible thermodynamics revisited (1988-98)

We summarize recent progress on one-dimensional and multidimensional stability of viscous shock wave solutions of compressible Navier–Stokes equations and related symmetrizable hyperbolic–parabolic systems, with an emphasis on the large-amplitude regime where transition from stability to instability may be expected to occur. The main result is the establishment of rigorous necessary and sufficient conditions for linearized and nonlinear planar viscous stability, agreeing in one dimension and separated in multidimensions by a co-dimension one set, that both extend and sharpen the formal conditions of structural and dynamical stability found in classical physical literature. The sufficient condition in multidimensions is new, and represents the main mathematical contribution of this article. The sufficient condition for stability is always satisfied for sufficiently small-amplitude shocks, while the necessary condition is known to fail under certain circumstances for sufficiently large-amplitude shocks; both are readily evaluable numerically. The precise conditions under and the nature in which transition from stability to instability occurs are outstanding open questions in the theory. 1

Stability of Large-Amplitude Shock Waves of Compressible Navier–Stokes Equations

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.

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

The stability of plane shock waves in Magnetohydrodynamics for an ideal medium is studied. Stability results are obtained for the special case of fast parallel shock waves in a polytropic gas. Linear stability is proved for a polytropic gas with arbitrary γ. The domain of structural (nonlinear) stability, where the uniform Lopatinsky condition is fulfilled for the stability problem, is found.

Stability of fast parallel MHD shock waves in polytropic gas

The stability of plane magnetohydrodynamic shock waves in a collisionless strongly magnetised plasma (described by Chew, Goldberger and Low equations) is studied. Stability results are obtained for the special cases of parallel and transversal fast shock waves by the test of fulfilment of the uniform Lopatinsky condition for corresponding linear stability problems. It is shown that fast parallel shock waves are always stable. The stability domain for fast transversal shock waves is found.

Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy

Well-posedness of a linear mixed problem on stability of electrohydrodynamic shock waves is investigated in the paper. Stability of shock waves for a hydrodynamic model of movement of a continuum with a volume electric charge is proved.

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Investigation on stability of electrohydrodynamic shock waves

Nous analysons la stabilite lineaire des ondes de choc dans un gaz relativiste radiatif. Les chocs radiatifs sont classifies comme chocs lents et rapides selon le rapport entre la vitesse du son et la vitesse du gaz a l'arriere et a l'avant du front de discontinuite. A l'aide de la technique des integrales d'energie dissipative on demontre que les ondes de choc rapides sont stables. On construit aussi un exemple mal pose pour le probleme de la stabilite des chocs lents.

Yu. L. Trakhinin

Papers

Stability of fast parallel MHD shock waves in polytropic gas

Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy

Investigation on stability of electrohydrodynamic shock waves

Stability of shock waves in relativistic radiation hydrodynamics

Rotational discontinuity in magnetohydrodynamics