×

Energy dispersed large data wave maps in \(2+1\) dimensions. (English) Zbl 1218.35129

The authors consider finite energy large data wave maps from the Minkowski space \(\mathbb R^{2+1}\) into a compact Riemannian manifold \((M,g)\).
The Cauchy problem for the wave map equation has the form
\[ (\partial_t^2-\Delta)\varphi^a=-S^a_{bc}(\varphi)\partial_\alpha \varphi^b \partial^\alpha \varphi^c, \quad\varphi\in\mathbb R^N, \]
with initial data \(\varphi(0,x)=\varphi_0(x)\), \(\partial_t \varphi(0,x)=\varphi_1(x)\) obeying the constraints \(\varphi_0(x)\in M\), \(\varphi_1(x)\in T_{\varphi_0(x)}M\), \(x\in\mathbb R^2\). The system admits a conserved energy,
\[ E[\varphi](t):=\int_{\mathbb R^2}(|\partial_t \varphi|^2+|\nabla_x \varphi|^2)\,dx=E. \]
The main result of the paper is a conditional regularity theorem, which states that there exist two functions \(0<\varepsilon(E)\ll 1\ll F(E)\) of the energy such that the following statement is true. If \(\varphi\) is a finite energy solution on the open interval \((t_1,t_2)\) with energy \(E\) and \(\sup_k\|P_k \varphi\|_{L^\infty_{t,x}[(t_1,t_2)\times\mathbb R^2]}\leq \varepsilon(E)\), then one also has \(\|\varphi\|_{S(t_1,t_2)}\leq F(E)\), and such a solution \(\varphi(t)\) extends in a regular way to a neighborhood of the interval \([t_1,t_2]\), where \(S\) is a variant of the standard dispersive norm from D. Tataru [Am. J. Math. 127, No. 2, 293–377 (2005; Zbl 1330.58021)]. The authors also prove a weaker non-conditional version of this result as a consequence of its frequency envelope version.

MSC:

35L15 Initial value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
58J45 Hyperbolic equations on manifolds

Citations:

Zbl 1330.58021
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Bahouri H., Gérard P.: High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math. 121(1), 131–175 (1999) · Zbl 0919.35089
[2] Bejenaru, I., Ionescu, A., Kenig, C., Tataru, D.: Global Schrödinger maps. http://arXiv.org/abs/0807.0265v1[math.AD] , 2008
[3] Gromov M.L.: Isometric imbeddings and immersions. Dokl. Akad. Nauk SSSR 192, 1206–1209 (1970) · Zbl 0214.50404
[4] Günther, M.: Isometric embeddings of Riemannian manifolds. In: Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Tokyo: Math. Soc. Japan, 1991, pp. 1137–1143 · Zbl 0745.53031
[5] Hadac M., Herr S.S., Koch H.: Well-posedness and scattering for the kp-ii equation in a critical space. Ann. I. H. Poicavé - AN 26(3), 917–941 (2009) · Zbl 1169.35372
[6] Kenig C.E., Merle F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008) · Zbl 1183.35202
[7] Klainerman S., Machedon M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46(9), 1221–1268 (1993) · Zbl 0803.35095
[8] Klainerman S., Rodnianski I.: On the global regularity of wave maps in the critical Sobolev norm. Internat. Math. Res. Notices 13, 655–677 (2001) · Zbl 0985.58009
[9] Klainerman S., Selberg S.: Remark on the optimal regularity for equations of wave maps type. Comm. Part. Diff. Eqs. 22(5-6), 901–918 (1997) · Zbl 0884.35102
[10] Klainerman S., Selberg S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 4(2), 223–295 (2002) · Zbl 1146.35389
[11] Koch H., Tataru D.: Dispersive estimates for principally normal pseudodifferential operators. Comm. Pure Appl. Math. 58(2), 217–284 (2005) · Zbl 1078.35143
[12] Koch, H., Tataru, D.: A priori bounds for the 1D cubic NLS in negative Sobolev spaces. Int. Math. Res. Not. IMRN, 16:Art. ID rnm053, 36, (2007) · Zbl 1169.35055
[13] Krieger J., Schlag W., Tataru D.: Renormalization and blow up for charge one equivariant critical wave maps. Invent. Math. 171(3), 543–615 (2008) · Zbl 1139.35021
[14] Krieger J.: Global regularity of wave maps from \({{\mathbb R}^ {3+1}}\) to surfaces. Commun. Math. Phys. 238(1-2), 333–366 (2003) · Zbl 1046.58010
[15] Krieger J.: Global regularity of wave maps from \({{\mathbb R}^ {2+1}}\) to H 2. Small energy. Commun. Math. Phys. 250(3), 507–580 (2004) · Zbl 1099.58010
[16] Krieger, J., Schlag, W.: Concentration compactness for critical wave-maps. http://arXiv.org/abs/0908.2974v1[math.AP] , 2009 · Zbl 1387.35006
[17] Nahmod A., Stefanov A., Uhlenbeck K.: On the well-posedness of the wave map problem in high dimensions. Comm. Anal. Geom. 11(1), 49–83 (2003) · Zbl 1085.58022
[18] Nash J.: The imbedding problem for Riemannian manifolds. Ann. of Math. (2) 63, 20–63 (1956) · Zbl 0070.38603
[19] Rodnianski, I., Sterbenz, J.: On the formation of singularities in the critical O(3) sigma-model. http://arXiv.org/abs/math/0605023v3[mth.AP] , 2008
[20] Shatah J., Struwe M.: The Cauchy problem for wave maps. Int. Math. Res. Not. 11, 555–571 (2002) · Zbl 1024.58014
[21] Sterbenz, J., Tataru, D.: Regularity of Wave-Maps in Dimension 2+1. doi: 10.1007/s00220-010-1062-3 · Zbl 1218.35057
[22] Tao, T.: Global regularity of wave maps III. Large energy from \({{\mathbb {R}}^ {2+1}}\) to hyperbolic spaces. http://arXiv.org/abs/0805.4666v3[math.AP] , 2009
[23] Tao, T.: Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class. http://arXiv.org/abs/0806.3592v2[math.AP] , 2009
[24] Tao, T.: Global regularity of wave maps V. Large data local well-posedness in the energy class. http://arXiv.org/abs/0808.0368v2[math.AP] , 2009
[25] Tao, T.: Global regulrity of wave maps VI. Abstract theory of Minimal energy blowup solutions. http://arXiv.org/abs/0906.2833v2[math.AP] , 2009
[26] Tao, T.: An inverse theorem for the bilinear L 2 Strichartz estimate for the wave equation. http://arXiv.org/abs/0904.2880v1[math.AP] , 2009
[27] Tao T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238(2), 215–268 (2001) · Zbl 0992.42004
[28] Tao T.: Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices 6, 299–328 (2001) · Zbl 0983.35080
[29] Tao T.: Global regularity of wave maps. II. Small energy in two dimensions. Commun. Math. Phys. 224(2), 443–544 (2001) · Zbl 1020.35046
[30] Tao, T.: Geometric renormalization of large energy wave maps. In: Journées ”Équations aux Dérivées Partielles”, pages Exp. No. XI, 32. École Polytech., Palaiseau, 2004
[31] Tataru D.: Local and global results for wave maps. I. Comm. Part. Diff. Eqs. 23(9-10), 1781–1793 (1998) · Zbl 0914.35083
[32] Tataru D.: On global existence and scattering for the wave maps equation. Amer. J. Math. 123(1), 37–77 (2001) · Zbl 0979.35100
[33] Tataru D.: Rough solutions for the wave maps equation. Amer. J. Math. 127(2), 293–377 (2005) · Zbl 1330.58021
[34] Wolff T.: A sharp bilinear cone restriction estimate. Ann. of Math. (2) 153(3), 661–698 (2001) · Zbl 1125.42302
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.