A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations

Alinhac, Serge

doi:10.5802/jedp.599

Cited by 32 publications

(64 citation statements)

References 15 publications

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“…(1) is a consequence of the finite speed of propagation and ODE techniques (see for example Levine [11] and Antonini and Merle [4]). More blow-up results can be found in Caffarelli and Friedman [5], Alinhac [1,2], Kichenassamy and Littman [9,10] and Shatah and Struwe [21]). Note that an important part of the literature on blow-up in the wave framework is devoted to quasilinear wave equations (where the nonlinearity occurs in the diffusion term).…”

Section: The Problem and Known Resultsmentioning

confidence: 87%

“…Note that an important part of the literature on blow-up in the wave framework is devoted to quasilinear wave equations (where the nonlinearity occurs in the diffusion term). Such equations may develop "geometric" blow-up (see Alinhac [1][2][3]). …”

Section: The Problem and Known Resultsmentioning

confidence: 98%

“…Since κ(d 2 , ·) = T d 2 κ 0 by (33), we see from Lemma 2.8 that for all d 2 ∈ (−1, 1), κ(d 2 , ·) H 0 κ 0 H 0 C. Therefore, κ(d 2 , ·) − κ 0 H 0 is a bounded C 1 function of θ 2 = 1 2 log( 1+d 2 1−d 2 ) which is zero when d 2 is zero. This directly implies (174).…”

Section: Proof Of (174)mentioning

confidence: 84%

“…By convergence in energy space, we obtain for all δ > 0 and s ∈ (− log T + 1, ∞), s+1 s |y|<1−δ (∂ y w) 2 1 − y 2 + |w| p+1 + w 2 + (∂ s w) 2 ρ C(E 0 + 1). which proves for all s − log T + 2, E(w(s)) 0 from (A.2).…”

Section: Appendix a Positivity Of The Lyapunov Functional E(w)mentioning

confidence: 96%

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Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

Merle

Zaag

2007

Journal of Functional Analysis

211

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In this paper, we consider the semilinear wave equation with a power nonlinearity in one space dimension. We exhibit a universal one-parameter family of functions which stand for the blow-up profile in self-similar variables at a non-characteristic point, for general initial data. The proof is done in self-similar variables. We first characterize all the solutions of the associated stationary problem, as a one parameter family. Then, we use energy arguments coupled with dispersive estimates to show that the solution approaches this family in the energy norm, in the non-characteristic case, and to a finite decoupled sum of such a solution in the characteristic case. Finally, in the case where this sum is reduced to one element, which is the case for noncharacteristic points, we use modulation theory coupled with a nonlinear argument to show the exponential convergence (in the self-similar time variable) of the various parameters and conclude the proof. This step provides us with a result of independent interest: the trapping of the solution in self-similar variables near the set of stationary solutions, valid also for non-characteristic points. The proof of these results is based on a new analysis in the self-similar variable.

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Section: The Problem and Known Resultsmentioning

confidence: 87%

Section: The Problem and Known Resultsmentioning

confidence: 98%

Section: Proof Of (174)mentioning

confidence: 84%

Section: Appendix a Positivity Of The Lyapunov Functional E(w)mentioning

confidence: 96%

See 2 more Smart Citations

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

Merle

Zaag

2007

Journal of Functional Analysis

211

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“…In this article we study the well-posedness properties of the Cauchy problem (1)- (2). In the inviscid case for ν = 0 , the Cauchy problem for the Kuznetsov equation is a particular case of a general quasi-linear hyperbolic system of the second order considered by Hughes, Kato and Marsden [8] (see Theorem 1 Points 1 and 2 for the application of their results to the Kuznetsov equation).…”

Section: Introductionmentioning

confidence: 99%

Cauchy problem for the Kuznetsov equation

Dekkers¹,

Rozanova-Pierrat²

2019

Discrete &Amp; Continuous Dynamical Systems - A

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We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2 -stability estimate, obtained between a regular and a less regular solutions.

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Supercritical Geometric Optics for Nonlinear Schrödinger Equations

Alazard

Carles

2008

Arch Rational Mech Anal

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We consider the small time semi-classical limit for nonlinear Schrödinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To overcome this issue, we introduce new unknown functions, which are defined nonlinearly in terms of the wave function itself. This approach provides a local version of the modulated energy functional introduced by Y. Brenier. The system we obtain is hyperbolic symmetric, and the justification of WKB analysis follows. Support by the ANR project SCASEN is acknowledged.

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A minicourse on global existence and blowup of classical solutions to multidimensional quasilinear wave equations

Cited by 32 publications

References 15 publications

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension

Cauchy problem for the Kuznetsov equation

Supercritical Geometric Optics for Nonlinear Schrödinger Equations

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