MR3052716 Sijia ZHONG Sijia ZHONG The Cauchy Problem of Null Form Wave Equation on $\mathbf{T}^d$ with Random Initial Data Funkcialaj Ekvacioj. Serio Internacia 55 2012 367--403 http://fe.math.kobe-u.ac.jp/FE/FullPapers/55-3/55_367.pdf http://www.ams.org/mathscinet-getitem?mr=MR3052716 In this paper, we consider the local well-posedness of the wave equation with null form nonlinearities under the periodic boundary condition. First, we prove the local existence based on some $L^p$-based Sobolev like Space $\hat{H}^{s,p}$, where $s>\frac{1}{2}+\frac{d}{p'}$ for $d\geq 2$, $s\geq\frac{1}{2}$, for $d=1$. Second, we prove the local existence for randomized initial data in $H^s$ with $s>\frac{1}{2}$ and $d\geq 2$. Wave equations, Null form, Random initial data. 46E35, 35L15, 58J45. 55-367 2012 The Cauchy Problem of Null Form Wave Equation on $\mathbf{T}^d$ with Random Initial Data Sijia ZHONG Sijia ZHONG
1 Bourgain, J. Refinements of Strichartz inequality and applications to 2D-NLS with critical nonlinearity Internat. Math. Res. Notices 1998 1998 253-283 MR1616917
2 Bournaveas, N. Local well-posedness for a nonlinear Dirac equation in spaces of almost critical dimension Discrete Contin. Dyn. Syst. 20 2008 605-616 MR2373206
3 Grünrock, A. Bi- and trilinear Schrödinger estimates in one space dimension with applications to cubic NLS and DNLS Int. Math. Res. Not. 2005 2005 2525-2558 MR2181058
4 Grünrock, A. An improved local well-posedness result for the modified KdV equation Int. Math. Res. Not. 2004 2004 3287-3308 MR2096258
5 Burq, N.; Planchon, F. Global existence for energy critical waves in 3-D domains: Neumann boundary conditions Amer. J. Math. 131 2009 1715-1742 MR2567505
6 Burq, N.; Tzvetkov, N. Invariant measure for the three dimensional nonlinear wave equation Int. Math. Res. Not. IMRN 2007 2007 Art. ID rnm108, 26 pp MR2376217
7 Burq, N.; Tzvetkov, N. Random data Cauchy theory for supercritical wave equations I: local theory Invent. Math. 173 2008 449-475 MR2425133
8 Burq, N.; Tzvetkov, N. Random data Cauchy theory for supercritical wave equations II: a global existence result Invent. Math. 173 2008 477-496 MR2425134
9 Ginibre, J. Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d'espace Séminaire Bourbaki, Vol. 1994/95, Astérisque No. 237 1996 Exp. No. 796, 4, 163-187 MR1423623
10 Ginibre, J.; Tsutsumi, Y.; Velo, G. On the Cauchy problem for the Zakharov system J. Funct. Anal. 151 1997 384-436 MR1491547
11 Keel, M.; Tao, T. Local and Global Well-posedness of Wave Maps on $\mathbb{R}^{1+1}$ for Rough Data Internat. Math. Res. Notices 1998 1998 1117-1156 MR1663216
12 Kenig, C. E.; Ponce, G.; Vega, L. Quadratic forms for the 1-D semilinear Schrödinger equation Trans. Amer. Math. Soc. 348 1996 3323-3353 MR1329387
13 Klainerman, S.; Machedon, M. Space-time estimates for null forms and the local existence theorem Comm. Pure Appl. Math. 46 1993 1221-1268 MR1231427
14 Klainerman, S.; Machedon, M. Smoothing estimates for null forms and applications Duke Math. J. 81 1995 99-133 MR1381973
15 Klainerman, S.; Selberg, S. Bilinear estimates and applications to nonlinear wave equations Commun. Contemp. Math. 4 2002 223-295 MR1901147
16 Ledoux, M.; Talagrand, M. Probability in Banach Space Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 23, Springer-Verlag, Berlin 1991 MR1102015
17 Lindblad, H. Counterexamples to local existence for quasilinear wave equations Math. Res. Lett. 5 1998 605-622 MR1666844
18 Nakanishi, K. Local wellposedness and illposedness in the critical Besov spaces for semilinear wave equations with quadratic forms Funkcial. Ekvac. 42 1999 261-279 MR1718767 1718767
19 Ponce, G.; Sideris, T. Local regularity of nonlinear wave equations in three space dimensions Comm. Partial Differential Equations 18 1993 169-177 MR1211729
20 Selberg, S. On an estimate for the wave equation and applications to nonlinear problem Differential Integral Equation 15 2002 213-236 MR1870470
21 Veraar, M. Regularity of Gaussian white noise on the $d$-dimensional torus Banach Center Publ., 95, Polish Acad. Sci. Inst. Math., Warsaw 2011 pp. 385-398 MR2918350