Displaying similar documents to “On Lars Hörmander’s remark on the characteristic Cauchy problem”

Global existence for coupled Klein-Gordon equations with different speeds

Pierre Germain (2011)

Annales de l’institut Fourier

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Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.

Periodic conservative solutions of the Camassa–Holm equation

Helge Holden, Xavier Raynaud (2008)

Annales de l’institut Fourier

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We show that the periodic Camassa–Holm equation u t - u x x t + 3 u u x - 2 u x u x x - u u x x x = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u | t = 0 in H per 1 . The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with μ ac = ( u 2 + u x 2 ) d x . The total energy is preserved by the solution.

L p - L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

Jerzy Gawinecki (1991)

Annales Polonici Mathematici

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We prove the L p - L q -time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the L p - L q -time decay estimates.

Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity

Boris Haspot (2012)

Annales de l’institut Fourier

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This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ℝ N with N ≥ 2 . We address the question of well-posedness for and initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where u 0 ∈ B p , r N p - 1 with 1 ≤ p < + ∞ , 1 ≤ r ≤ + ∞ ). This improves the classical analysis where u 0 is considered belonging in B p , 1 N p - 1 such that the velocity u remains...

Invariant measures for the defocusing Nonlinear Schrödinger equation

Nikolay Tzvetkov (2008)

Annales de l’institut Fourier

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We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schrödinger equations on the disc of the plane ℝ 2 . We also prove an estimate giving some intuition to what may happen in 3 dimensions.

Global existence for a quasilinear wave equation outside of star-shaped domains

Hart F. Smith (2001)

Journées équations aux dérivées partielles

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This talk describes joint work with Chris Sogge and Markus Keel, in which we establish a global existence theorem for null-type quasilinear wave equations in three space dimensions, where we impose Dirichlet conditions on a smooth, compact star-shaped obstacle 𝒦 ⊂ ℝ 3 . The key tool, following Christodoulou [1], is to use the Penrose compactification of Minkowski space. In the case under consideration, this reduces matters to a local existence theorem for a singular obstacle problem. Full details...