This talk gives a brief review of some recent progress in the asymptotic analysis of short pulse solutions of nonlinear hyperbolic partial differential equations. This includes descriptions on the scales of geometric optics and diffractive geometric optics, and also studies of special situations where pulses passing through focal points can be analysed.
For the dynamics , an equilibrium point are always unstable when on a neighborhood of the non constant satisfies for a real second order elliptic . The proof uses a result of Kozlov [6].
Pour tout réel positif , on étudie la propagation de la régularité locale pour des solutions d’équations aux dérivées partielles hyperboliques non linéaires, admettant a priori la régularité minimale permettant de définir les expressions non linéaires figurant dans l’équation. En particulier, on démontre le théorème de propagation dans le cas des solutions essentiellement bornées (resp. lipschitziennes) de systèmes du premier ordre semi-linéaires (resp. quasi-linéaires).
We study spherical pulse like families of solutions to semilinear wave equattions in space time of dimension 1+3 as the pulses focus at a point and emerge outgoing. We emphasize the scales for which the incoming and outgoing waves behave linearly but the nonlinearity has a strong effect at the focus. The focus crossing is described by a scattering operator for the semilinear equation, which broadens the pulses. The relative errors in our approximate solutions are small in the L norm.
Semilinear hyperbolic problems with source terms piecewise smooth and discontinuous across characteristic surfaces yield similarly piecewise smooth solutions. If the discontinuous source is replaced with a smooth transition layer, the discontinuity of the solution is replaced by a smooth internal layer. In this paper we describe how the layer structure of the solution can be computed from the layer structure of the source in the limit of thin layers. The key idea is to use a transmission problem...
We analyse Bérenger’s split algorithm applied to the system version of the two dimensional wave equation with absorptions equal to Heaviside functions of , . The methods form the core of the analysis [11] for three dimensional Maxwell equations with absorptions not necessarily piecewise constant. The split problem is well posed, has no loss of derivatives (for divergence free data in the case of Maxwell), and is perfectly matched.
We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.
We study oscillatory solutions of semilinear first order symmetric hyperbolic system , with real analytic . The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in with only the natural hypothesis of coherence. In the special case where has constant coefficients and the phases are linear, the solutions have asymptotic description where the profile is almost periodic in . ...
The nonlinear dissipative wave equation in dimension has strong solutions with the following structure. In the solutions have a focusing wave of singularity on the incoming light cone . In that is after the focusing time, they are smoother than they were in . The examples are radial and piecewise smooth in
If is a linear hyperbolic system of partial differential operators for which local uniqueness in the Cauchy problem at spacelike hypersurfaces is known, we find nearly optimal domains of determinacy of open sets . The frozen constant coefficient operators determine local convex propagation cones, . Influence curves are curves whose tangent always lies in these cones. We prove that the set of points which cannot be reached by influence curves beginning in the exterior of is a domain of...
2000 Mathematics Subject Classification: 35L05, 35P25, 47A40. The problem studied here was suggested to us by V. Petkov. Since the beginning of our careers, we have benefitted from his insights in partial differential equations and mathematical physics. In his writings and many discussions, the conjuction of deep analysis and specially interesting problems has been a source inspiration for us. The research of J. Rauch is partially supported by the U.S. National Science Foundation under...
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