Random Data Cauchy Theory for Nonlinear Wave Equations of Power-Type on ℝ³

References

• Bahouri , H. , Chemin , J. ( 2006 ). On global well-posedness for defocusing cubic wave equation . Int. Math. Res. Not. 2006 : 1 – 12 .
   Web of Science ®Google Scholar
• Bényi , Á. , Oh , T. , Pocovnicu , O. On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on ℝ d , d ≥ 3. Preprint, arXiv:1405.7327.
   Google Scholar
• Bényi , Á. , Oh , T. , Pocovnicu , O. Wiener randomization on unbounded domains and an application to almost sure well-posedness of NLS. Preprint, arXiv:1405.7326.
   Google Scholar
• Bourgain , J. ( 1994 ). Periodic nonlinear Schrödinger equation and invariant measures . Comm. Math. Phys. 166 : 1 – 26 .
   Web of Science ®Google Scholar
• Bourgain , J. ( 1996 ). Invariant measures for the 2D-defocusing nonlinear Schrödinger equation . Comm. Math. Phys. 176 : 421 – 445 .
   Web of Science ®Google Scholar
• Bourgain , J. (1998). Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int. Math. Res. Not. 1998:253–283.
   Web of Science ®Google Scholar
• Bourgain , J. , Bulut , A. ( 2014 ). Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3d ball . J. Funct. Anal. 266 : 2319 – 2340 .
   Web of Science ®Google Scholar
• Burq , N. , Thomann , L. , Tzvetkov , N. Long time dynamics for the one dimensional non linear Schrödinger equation. Preprint, arXiv:1002.4054.
   Google Scholar
• Burq , N. , Tzvetkov , N. ( 2007 ). Invariant measure for a three dimensional nonlinear wave equation . Int. Math. Res. Not. 2007 : 1 – 26 .
   Web of Science ®Google Scholar
• Burq , N. , Tzvetkov , N. ( 2008 ). Random data Cauchy theory for supercritical wave equations. I. Local theory . Invent. Math. 173 : 449 – 475 .
   Web of Science ®Google Scholar
• Burq , N. , Tzvetkov , N. ( 2008 ). Random data Cauchy theory for supercritical wave equations. II. A global existence result . Invent. Math. 173 : 477 – 496 .
   Web of Science ®Google Scholar
• Burq , N. , Tzvetkov , N. ( 2014 ). Probabilistic well-posedness for the cubic wave equation . J. Eur. Math. Soc. 16 ( 1 ): 1 – 30 .
   Web of Science ®Google Scholar
• Christ , M. , Colliander , J. , Tao , T. Ill-posedness for nonlinear Schrödinger and wave equations. Preprint, arXiv:math/0311048.
   Google Scholar
• Colliander , J. , Oh , T. ( 2012 ). Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L 2(𝕋) . Duke Math. J. 161 : 367 – 414 .
   Web of Science ®Google Scholar
• Deng , Y. ( 2012 ). Two-dimensional nonlinear Schrödinger equation with random radial data . Anal. PDE 5 : 913 – 960 .
   Web of Science ®Google Scholar
• Gallagher , I. , Planchon , F. ( 2003 ). On global solutions to a defocusing semi-linear wave equation . Rev. Mat. Iberoamericana 19 : 161 – 177 .
   Web of Science ®Google Scholar
• Ginibre , J. , Velo , G. ( 1989 ). The global Cauchy problem for the nonlinear Klein-Gordon equation. II . Ann. Inst. H. Poincaré Anal. Non Linéaire 6 : 15 – 35 .
   Web of Science ®Google Scholar
• Ginibre , J. , Velo , G. ( 1995 ). Generalized Strichartz inequalities for the wave equation . J. Funct. Anal. 133 : 50 – 68 .
   Web of Science ®Google Scholar
• Ibrahim , S. , Majdoub , M. , Masmoudi , N. ( 2007 ). Ill-posedness of H 1-supercritical waves . C. R. Math. Acad. Sci. Paris 345 : 133 – 138 .
   Web of Science ®Google Scholar
• Keel , M. , Tao , T. ( 1998 ). Endpoint Strichartz estimates . Amer. J. Math. 120 : 955 – 980 .
   Web of Science ®Google Scholar
• Kenig , C. , Ponce , G. , Vega , L. ( 2000 ). Global well-posedness for semi-linear wave equations . Comm. Part. Diff. Eqs. 25 : 1741 – 1752 .
   Web of Science ®Google Scholar
• Lebeau , G. ( 2005 ). Perte de régularité pour les équations d'ondes sur-critiques [Loss of regularity for supercritical wave equations] . Bull. Soc. Math. France 133 : 145 – 157 .
   Web of Science ®Google Scholar
• Lebowitz , J. , Rose , H. , Speer , E. ( 1988 ). Statistical mechanics of the nonlinear Schrödinger equation . J. Statist. Phys. 50 : 657 – 687 .
   Web of Science ®Google Scholar
• Lindblad , H. , Sogge , C. ( 1995 ). On existence and scattering with minimal regularity for semilinear wave equations . J. Funct. Anal. 130 : 357 – 426 .
   Web of Science ®Google Scholar
• Nahmod , A. , Oh , T. , Rey-Bellet , L. , Staffilani , G. ( 2012 ). Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS . J. Eur. Math. Soc. 14 : 1275 – 1330 .
   Web of Science ®Google Scholar
• Nahmod , A. , Pavlović , N. , Staffilani , G. ( 2013 ). Almost sure existence of global weak solutions for supercritical Navier-Stokes equations . SIAM J. Math. Anal. 45 : 3431 – 3452 .
   Web of Science ®Google Scholar
• Nahmod , A. , Rey-Bellet , L. , Sheffield , S. , Staffilani , G. ( 2011 ). Absolute continuity of Brownian bridges under certain gauge transformations . Math. Res. Lett. 18 : 875 – 887 .
   Web of Science ®Google Scholar
• Oh , T. (2009). Invariance of the white noise for KdV. Comm. Math. Phys. 292:217–236.
   Web of Science ®Google Scholar
• Pecher , H. ( 1984 ). Nonlinear small data scattering for the wave and Klein-Gordon equation . Math. Z. 185 : 261 – 270 .
   Web of Science ®Google Scholar
• Poiret , A. Solutions globales pour des équations de Schrödinger sur-critiques en toutes dimensions [Global solutions for supercritical Schrödinger equations in all dimensions]. Preprint, arXiv:1207.3519.
   Google Scholar
• Poiret , A. Solutions globales pour l’équation de Schrödinger cubique en dimension 3 [Global solutions for the cubic Schrödinger equation in dimension 3]. Preprint, arXiv:1207.1578.
   Google Scholar
• Poiret , A. , Robert , D. , Thomann , L. Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. Analysis & PDE, to appear.
   Google Scholar
• Richards , G. Invariance of the gibbs measure for the periodic quartic gkdv. Preprint, arXiv:1209.4337.
   Google Scholar
• Roy , T. ( 2009 ). Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on ℝ3 . Discrete Contin. Dyn. Syst. 24 : 1307 – 1323 .
   Web of Science ®Google Scholar
• de Suzzoni , A. ( 2013 ). Large data low regularity scattering results for the wave equation on the Euclidean space . Commun. Partial Differ. Equations 38 : 1 – 49 .
   Web of Science ®Google Scholar
• de Suzzoni , A. ( 2014 ). Consequences of the choice of a particular basis of L 2(S 3) for the cubic wave equation on the sphere and the Euclidean space . Commun. Pure Appl. Anal. 13 : 991 – 1015 .
   Web of Science ®Google Scholar
• Strichartz , R. ( 1977 ). Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations . Duke Math. J. 44 : 705 – 714 .
   Web of Science ®Google Scholar
• Thomann , L. , Tzvetkov , N. ( 2010 ). Gibbs measure for the periodic derivative nonlinear Schrödinger equation . Nonlinearity 23 : 2771 – 2791 .
   Web of Science ®Google Scholar
• Tzvetkov , N. ( 2008 ). Invariant measures for the defocusing nonlinear Schrödinger equation . Ann. Inst. Fourier (Grenoble) 58 : 2543 – 2604 .
   Web of Science ®Google Scholar
• Tzvetkov , N. ( 2010 ). Construction of a Gibbs measure associated to the periodic Benjamin-Ono equation . Probab. Theory Related Fields 146 : 481 – 514 .
   Web of Science ®Google Scholar
• Zhang , T. , Fang , D. ( 2012 ). Random data Cauchy theory for the generalized incompressible Navier-Stokes equations . J. Math. Fluid Mech. 14 : 311 – 324 .
   Web of Science ®Google Scholar