Oblique magnetohydrodynamic shock wave decay

Summary A modified hodograph transformation is used to obtain an exact solution of the equations governing the one-dimensional unsteady flow of an ideal, inviscid, perfectly conducting compressible fluid, subjected to an oblique magnetic field. This solution is used to obtain an approximate representation of the path of an initially uniform shock wave which intersects a centered simple wave. The solutions for the corresponding problems in the conventional, non-magnetic case and for a transverse orientation of the applied magnetic field are contained as special limiting cases of the solutions of the present paper. This provides a valuable check on the theory.

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Résumé Une transformation hodographe modifiée est employée pour obtenir une solution exacte des équations relatives aux écoulements unidimensionnels non-stationnaires et non-isentropiques d'un fluide non visqueux idéal, parfaitement conducteur d'électricité et compressible, soumis à l'action d'un champ magnétique oblique avec deux composantes différentes de zero. On utilise cette solution pour obtenir une représentation approximative de la trajectoire d'une onde de choc magnétohydrodynamique initialement uniforme, rencontrant une onde simple centrée. Les solutions pour le cas non magnétique et pour un champ transversal apparaissent comme des cas limites particuliers de la solution présentée ici, c'est là une confirmation de la validité de la theorie.

     

The decay of a magnetohydrodynamic shock wave

Résumé On utilise l'analyse linéaire pour étudier les faibles perturbations non-isentropiques des équations relatives aux écoulements unidimensionnels et non-stationnaires d'un fluide non visqueux idéal, parfaitement conducteur de l'électricité et compressible, soumis à l'action d'un champ magnétique transversal. On utilise la solution générale de la perturbation non-isentropique d'un écoulement par ondes simples centrées pour déterminer la perturbation qui se manifeste lorsqu'une onde de choc magnétohydrodynamique, tout d'abord uniforme et de force arbitraire, rencontre le régime d'onde simple.Dans le cas limité d'un champ magnétique nul, la solution se réduit exactement à celle du problème correspondant de la dynamique classique des gaz. C'est la une confirmation de la validité de la théorie.

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This research was supported by National Science Foundation grant GP 87190.     

Magnetohydrodynamic shock propagation in non-uniform ducts

Zusammenfassung Es wird die Störung bestimmt, die ein gleichförmiger magnetohydrodynamischer Stoss beliebiger Stärke durch eine Querschnittsveränderung der Strömung erleidet. Beim Durchgang durch die Querschnittsveränderung ändert sich die Stärke des Stosses, und die Strömung bleibt nicht länger isentropisch. Es ergeben sich zwei verschiedene Störungsanteile, nämlich ein stationärer, der unmittelbar durch die Querschnittsveränderung hervorgerufen wird, und ein zeitlich veränderlicher Anteil, der von der Reflektion des ersten Anteils an dem Stoss herrührt.Die Untersuchungen, die auf ein einatomiges Gas beschränkt sind, enthalten gewisse bekannte gasdynamische Ergebnisse als Spezialfälle.

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Sponsored by the United States Army under Contract no. DA-11-022-ORD-2059.     

Formation and decay of electromagnetic shock waves

Résumé Ce traité se rapporte principalement à la dérivation de la loi de détérioration asymptotique pour les ondes de choc électromagnetiques faibles se propageant dans un milieu de type ferromagnétique. On dérive la loi de détérioration pour les paramètres de choc en utilisant une, technique de corrélation entre une des solutions d'une onde simple et les équations d'un choc faible.     

Exponential decay for a neutral wave equation

A strongly damped wave equation involving a delay of neutral type in its second order derivative is considered. It is proved that solutions decay to zero exponentially despite the fact that delays are, in general, sources of instability.     

Perturbation of a shock wave

The Cauchy problem is considered for the perturbed Hopf equation u_t+uu_x=εf(u), ε→0. The solution in the continuity domain can be expanded in the standard asymptotic series in integral powers of the small parameter. An asymptotic representation is found for the line of propagation of the shock wave. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 462–466, March, 1999.     

Optimal decay rates for the viscoelastic wave equation

In this paper, we consider a viscoelastic equation with minimal conditions on the relaxation function g, namely, , where H is an increasing and convex function near the origin and ξ is a nonincreasing function. With only these very general assumptions on the behavior of gat infinity, we establish optimal explicit and general energy decay results from which we can recover the optimal exponential and polynomial rates when H(s)=s^p and p covers the full admissible range [1,2). We get the best decay rates expected under this level of generality, and our new results substantially improve several earlier related results in the literature.     

Weighted energy decay for 1D wave equation

We obtain a dispersive long time decay in weighted energy norms for solutions to the 1D wave equation with generic potential. The decay extends the results obtained by Murata for the 1D Schrödinger equation.     

Polynomial decay rate for the dissipative wave equation

Using Fourier integral operators with special amplitude functions, we analyze the stabilization of the wave equation in a three-dimensional bounded domain on which exists a trapped ray bouncing up and down infinitely between two parallel parts of the boundary.     

Local energy decay for the damped wave equation

We prove local energy decay for the damped wave equation on R^d${\mathbb{R}}^d$. The problem which we consider is given by a long range metric perturbation of the Euclidean Laplacian with a short range absorption index. Under a geometric control assumption on the dissipation we obtain an almost optimal polynomial decay for the energy in suitable weighted spaces. The proof relies on uniform estimates for the corresponding “resolvent”, both for low and high frequencies. These estimates are given by an improved dissipative version of Mourre's commutators method.     

Superexponential decay of solutions of a semilinear wave equation

Consider a smooth solution of $\text{u}{}_{\mathrm{tt}}\text{? Δu + q(x) ¦ u ¦}{}^{\mathrm{p?1}}\text{u = 0 x ? R}{}^3\text{, q ? 0}$ and is C¹, and 1 < p < 5. Assume that the initial data decay sufficiently rapidly at infinity, $\text{q(x) ? a}\text{exp}\text{(?b ¦ x ¦}{}^{\mathrm{c}}\text{), a, b > 0, c > 1}$, and for simplicity, q_r ? 0. Then the local energy decays faster than exponentially.     

Exponential energy decay of solutions of viscoelastic wave equations

In this paper we consider the nonlinear viscoelastic equation

     

Unsteady Magnetohydrodynamic Flows

It is easy enough to deduce from the exact solution for a basicunsteady magnetohydrodynamic channel flow that there are twotime scales present. The possible existence of two time scalesenables one to formulate a method of constructing approximatesolutions for several flow problems. The approximation schemeemployed here has the advantage over the more usual methodsof boundary layer analysis in that it yields for each physicalquantity a single representation which is valid for all times.Moreover, it is shown that the error involved in using thisapproximate solution is suitably small.     

Weak asymptotics of shock wave formation process

We construct an asymptotic (in a weak sense) solution corresponding to the shock wave formation in a special situation.     

Local energy decay for linear wave equations with variable coefficients

A uniform local energy decay result is derived to the linear wave equation with spatial variable coefficients. We deal with this equation in an exterior domain with a star-shaped complement. Our advantage is that we do not assume any compactness of the support on the initial data, and its proof is quite simple. This generalizes a previous famous result due to Morawetz [The decay of solutions of the exterior initial-boundary value problem for the wave equation, Comm. Pure Appl. Math. 14 (1961) 561-568]. In order to prove local energy decay, we mainly apply two types of ideas due to Ikehata-Matsuyama [L²-behaviour of solutions to the linear heat and wave equations in exterior domains, Sci. Math. Japon. 55 (2002) 33-42] and Todorova-Yordanov [Critical exponent for a nonlinear wave equation with damping, J. Differential Equations 174 (2001) 464-489].     

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially even if m > 2.     

Time decay for nonlinear wave equations in two space dimensions

The Cauchy Problem for the equation u_tt–u+|u|^p–1u=0 (x², t>0, >1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For >5, it is shown that the maximum amplitude of such a wave decays at the expected rate t^–1/2 as t. For 1+8<5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when >_o, where _o is the root of the cubic equation ³-2²-7-8=0; thus _o4.15.Alfred P. Sloan Research Fellow     

decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: (□+∂_t)u=0 in with (u,∂_tu)|_t=0=(u₀,u₁). Because the L² estimates and the L^∞ estimates of the solution u(t) are well known, in this paper we pay attention to the L^p estimates with 1p<2 (in particular, p=1) of the solution u(t) for t0. In order to derive L^p estimates we first give the representation formulas of the solution u(t)=∂_tS(t)u₀+S(t)(u₀+u₁) and then we directly estimate the exact solution S(t)g and its derivative ∂_tS(t)g of the dissipative wave equation with the initial data (u₀,u₁)=(0,g). In particular, when p=1 and n1, we get the L¹ estimate: u(t)_L¹Ce^−t/4(u_0W^n,1+u_1W^n−1,1)+C(u_0L¹+u_1L¹) for t0.