2013/05/17 by Thomas Alazard, Alazard, Thomas, Jean-Marc Delort +1 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1305.4090
This paper is devoted to the proof of a global existence result for the water\nwaves equation with smooth, small, and decaying at infinity Cauchy data. We\nobtain moreover an asymptotic description in physical coordinates of the\nsolution, which shows that modified scattering holds.\n The proof is based on a bootstrap argument involving L2 and L^\∞\nestimates. The L2 bounds are proved in a companion paper of this article.\nThey rely on a normal forms paradifferential method allowing one to obtain\nenergy estimates on the Eulerian formulation of the water waves equation. We\ngive here the proof of the uniform bounds, interpreting the equation in a\nsemi-classical way, and combining Klainerman vector fields with the description\nof the solution in terms of semi-classical lagrangian distributions. This,\ntogether with the L2 estimates of the companion paper, allows us to deduce\nour main global existence result.\n