Jean-Luc Joly

Focusing at a point and absorption of nonlinear oscillations

Several recent papers give rigorous justifications of weakly nonlinear geometric optics. All of them consider oscillating wave trains on domains where focusing phenomena do not exist, either because the space dimension is equal to one, or thanks to a coherence assumption on the phases. This paper is devoted to a study of some nonlinear effects of focusing. In a previous paper, the authors have given a variety of examples which show how focusing in nonlinear equations can spoil even local existence in the sense that the domain of existence shrinks to zero as the wavelength decreases to zero. On the other hand, there are many problems for which global existence is known and in those cases it is natural to ask what happens to oscillations as they pass through a focus. The main goal of this paper is to present such a study for some strongly dissipative semilinear wave equations and spherical wavefronts which focus at the origin. We show that the strongly nonlinear phenomenon which is produced is that oscillations are killed by the simultaneous action of focusing and dissipation. Our study relies on the analysis of Young measures and two-scale Young measures associated to sequences of solutions. The main step is to prove that these measures satisfy appropriate transport equations. Then, their variances are shown to satisfy differential inequalities which imply a propagation result for their support.

Global Solutions to Maxwell Equations in a Ferromagnetic Medium

Abstract. We study the Cauchy problem for the Landau-Lifschitz model in ferromagnetism without exchange energy. Once existence of global finite energy solutions is obtained, we study additional uniqueness and regularity properties of these solutions.

A nonlinear instability for 3×3 systems of conservation laws

AbstractThe phenomenon of nonlinear resonance provides a mechanism for the unbounded amplification of small solutions of systems of conservation laws. We construct spatially 2π-periodic solutionsuN ∈C∞ ([0,tN] × ∝ withtN bounded, satisfying

A propos de l'existence et de la régularité des solutions de certaines inéquations quasi-variationnelles

\begin{array}{*{20}c} {\left\| {u^N } \right\|L^\infty ([0,t_N ] \times \mathbb{R}) \to 0,} & {\int\limits_0^{2\pi } {\left| {\partial _x u^N (0,x)} \right|dx \leqq C,} } \\ {\int\limits_0^{2\pi } {\left| {\partial _x u^N (t_N ,x)} \right|dx \geqq N, \left\| {u^N (t_N ,x)} \right\|L^p (\mathbb{R}) \geqq N\left\| {u^N (0,x)} \right\|L^p (\mathbb{R})} } & {1 \leqq p \leqq \infty .} \\ \end{array}

Global Solvability of the Anharmonic Oscillator Model from Nonlinear Optics

The variation grows arbitrarily large, and the sup norm is amplified by arbitrarily large factors.

HYPERBOLIC DOMAINS OF DETERMINACY AND HAMILTON–JACOBI EQUATIONS

For semilinear strictly hyperbolic systems Lu=f(x,u), we construct and justify high frequency nonlinear asymptotic expansions of the form u e (x)∼∑e j U j (x,φ(x)/e), Lu e −f(x,u e )∼0. The study of the principal term of such expansions is called nonlinear geometric optics in the applied literature. We show : (1) formal expansions with periodic profiles U j can be computed to all orders, (ii) the equations for the profiles from (i) are solvable, and (iii) there are solutions of the exact equations which have the formal series as high frequency asymptotic expansion

Several recent results in nonlinear geometric optics

Abstract We prove existence and regularity results for a general inequality including fixed point problems, variational and quasi variational inequalities. Examples of such problems are considered.

Recent Results in Non Linear Geometric Optics

Consider in ℝ1+d the semilinear wave equation