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Geometric optics and instability for NLS and Davey-Stewartson models

Rémi Carles, Eric Dumas, Christof Sparber (2012)

Journal of the European Mathematical Society

We study the interaction of (slowly modulated) high frequency waves for multi-dimensional nonlinear Schrödinger equations with Gauge invariant power-law nonlinearities and nonlocal perturbations. The model includes the Davey-Stewartson system in its elliptic-elliptic and hyperbolic-elliptic variants. Our analysis reveals a new localization phenomenon for nonlocal perturbations in the high frequency regime and allows us to infer strong instability results on the Cauchy problem in negative order Sobolev...

Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

Jean-François Coulombel, Olivier Guès (2010)

Annales de l’institut Fourier

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the source terms...

Global classical solutions to a kind of mixed initial-boundary value problem for inhomogeneous quasilinear hyperbolic systems

Yong-Fu Yang (2012)

Applications of Mathematics

In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant { ( t , x ) : t ≥ 0 , x ≥ 0 } is investigated. Under the assumption that the right-hand side satisfies a matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence and uniqueness of a C 1 solution and its L 1 stability with certain small initial and boundary data.

Global existence and stability of some semilinear problems

Mokhtar Kirane, Nasser-eddine Tatar (2000)

Archivum Mathematicum

We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.

Global in Time Stability of Steady Shocks in Nozzles

Jeffrey Rauch, Chunjing Xie, Zhouping Xin (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

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.

Global regularity for the 3D MHD system with damping

Zujin Zhang, Xian Yang (2016)

Colloquium Mathematicae

We study the Cauchy problem for the 3D MHD system with damping terms ε | u | α - 1 u and δ | b | β - 1 b (ε, δ > 0 and α, β ≥ 1), and show that the strong solution exists globally for any α, β > 3. This improves the previous results significantly.

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