Geometric renormalization of large energy wave maps

Tao, Terence

Journées équations aux dérivées partielles (2004), article no. 11, 32 p.

There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of “non-concentration” type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar.

[1] B.K. Berger, P. T. Chrusciel, V. Moncrief, On “asymptotically flat” spacetimes with $G_2$ invariant Cauchy surfaces, Ann. Physics 237 (1995), 322–354. | MR | Zbl

[2] P. Bizon, T. Chmaj, Z. Tabor, Formation of singularities for equivariant 2+1 dimensional wave maps into two-sphere, Nonlinearity 14 (2001), no. 5, 1041–1053. | MR | Zbl

[3] J. Bourgain, Global well-posedness of defocusing 3D critical NLS in the radial case, JAMS 12 (1999), 145-171. | MR | Zbl

[4] J. Bourgain, New global well-posedness results for non-linear Schrödinger equations, AMS Publications, 1999.

[5] T. Cazenave, J. Shatah, A.S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincaré Phys. Théor.68 (1998), 315-349. | Numdam | MR | Zbl

[6] Y. Choquet-Bruhat, Global existence theorems for hyperbolic harmonic maps, Ann. Inst. H. Poincare Phys. Theor. 46 (1987), 97–111. | Numdam | MR | Zbl

[7] D. Christodoulou, A. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math, 46 (1993), 1041–1091. | MR | Zbl

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering in the energy space for the critical nonlinear Schrödinger equation in ${ℝ}^3$, preprint.

[9] P. D’Ancona, V. Georgiev, On the continuity of the solution operator of the wave maps system, preprint.

[10] J. Eells, H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. | MR | Zbl

[11] A. Freire, S. Müller, M. Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. H. Poincare Anal. Non Lineaire 15 (1998), no. 6, 725–754. | Numdam | MR | Zbl

[12] J. Ginibre, G. Velo, The Cauchy problem for the $O\left(N\right)$, $CP(N-1)$, and $GC(N,P)$ models, Ann. Physics, 142 (1982), 393–415. | MR | Zbl

[13] L. Grafakos, R. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (1992), 124–164. | MR | Zbl

[14] M. Grillakis, On nonlinear Schrödinger equations. , Comm. Partial Differential Equations 25 (2000), no. 9-10, 1827–1844. | MR | Zbl

[15] M. Grillakis, Classical solutions for the equivariant wave map in $1+2$ dimensions, to appear in Indiana Univ. Math. J.

[16] C. Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math., 33,(1980), 727–737. | MR | Zbl

[17] F. Helein, Regularite des applications faiblement harmoniques entre une sur face et une varitee Riemannienne, C.R. Acad. Sci. Paris Ser. I Math., 312 (1991), 591-596. | MR | Zbl

[18] J. Isenberg, S. Liebling, Singularity formation in $2+1$ wave maps, J. Math. Phys. 43 (2002), no. 1, 678–683. | MR | Zbl

[19] M. Keel, Global existence for critical power Yang-Mills-Higgs equations in ${ℝ}^{3+1}$, Commun. in PDE 22 (1997), 1167–1227. | MR | Zbl

[20] M. Keel, T. Tao, Endpoint Strichartz Estimates, Amer. Math. J. 120 (1998), 955–980. | MR | Zbl

[21] M. Keel, T. Tao, Local and global well-posedness of wave maps on ${ℝ}^{1+1}$ for rough data, IMRN 21 (1998), 1117–1156. | MR | Zbl

[22] C. Kenig, E. Stein, Multilinear estimates and fractional interpolation, Math. Res. Lett. 6 (1999), 1–15 | MR | Zbl

[23] S. Klainerman, On the regularity of classical field theories in Minkowski space-time ${ℝ}^{3+1}$, Prog. in Nonlin. Diff. Eq. and their Applic., 29, (1997), Birkhäuser, 113–150. | MR | Zbl

[24] S. Klainerman, PDE as a unified subject, preprint.

[25] S. Klainerman, M. Machedon, Smoothing estimates for null forms and applications, Duke Math. J., 81 (1995), 99–133. | MR | Zbl

[26] S. Klainerman, M. Machedon, On the optimal local regularity for gauge field theories, Diff. and Integral Eq. 10 (1997), 1019–1030. | MR | Zbl

[27] S. Klainerman, I. Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, IMRN 13 (2001), 656–677. | MR | Zbl

[28] S. Klainerman, I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, preprint. | MR | Zbl

[29] S. Klainerman, S. Selberg, Remark on the optimal regularity for equations of wave maps type, C.P.D.E., 22 (1997), 901–918. | MR | Zbl

[30] S. Klainerman, S. Selberg, Bilinear estimates and applications to nonlinear wave equations, preprint. | Zbl

[31] J. Krieger, Global regularity of wave maps from ${ℝ}^{3+1}$ to ${\mathbf{H}}^2$, CMP 238 (2003), 333–366. | MR | Zbl

[32] J. Krieger, Global regularity of wave maps from ${ℝ}^{2+1}$ to ${\mathbf{H}}^2$. Small energy, preprint. | Zbl

[33] O.A. Ladyzhenskaya, V.I. Shubov, Unique solvability of the Cauchy problem for the equations of the two dimensional chiral fields, taking values in complete Riemann manifolds, J. Soviet Math., 25 (1984), 855–864. (English Trans. of 1981 Article.) | Zbl

[34] P. Li, L. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105 (1991), no. 1, 1–46. | MR | Zbl

[35] G. Liao, L. Tam, On the heat equation for harmonic maps from noncompact manifolds, Pacific J. Math. 153 (1992), no. 1, 129–145. | MR | Zbl

[36] S. Müller, M. Struwe, Global existence of wave maps in $1+2$ dimensions with finite energy data, Topol. Methods Nonlinear Anal. 7 (1996), 245–259. | MR | Zbl

[37] C. Muscalu, T. Tao, C. Thiele, Multilinear operators given by singular multipliers, J. Amer. Math. Soc. 15 (2002), 469–496. | MR | Zbl

[38] A. Nahmod, A. Stefanov, K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom. 11 (2003), 49–83. | MR | Zbl

[39] K. Nakanishi, Local well-posedness and Illposedness in the critical Besov spaces for semilinear wave equations with quadratic forms, Funk. Ekvac. 42 (1999), 261-279. | MR | Zbl

[40] K. Nakanishi, Energy scattering for non-linear Klein-Gordon and Schrodinger equations in spatial dimensions 1 and 2, JFA 169 (1999), 201–225. | MR | Zbl

[41] D. Oberlin, E. Stein, Mapping properties of the Radon transform, Indiana U. Math. J. 31 (1982), 641–650. | MR | Zbl

[42] K. Pohlmeyer, Integrable Hamiltonian systems and interaction through quadratic constraints, Comm. Math. Phys., 46 (1976), 207–221. | MR | Zbl

[43] R. Schoen, Analytic aspects of the harmonic map problem, Math. Sci. Res. Inst. Publ. 2 (1984), Springer, Berlin, 321–358. | MR | Zbl

[44] R. Schoen, S.T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Comment. Math. Helv. 51 (1976), no. 3, 333–341. | MR | Zbl

[45] S. Selberg, Multilinear space-time estimates and applications to local existence theory for non-linear wave equations, Princeton University Thesis.

[46] J. Shatah, Weak solutions and development of singularities of the $SU\left(2\right)$ $\sigma$-model. Comm. Pure Appl. Math., 41 (1988), 459–469. | MR | Zbl

[47] J. Shatah, The Cauchy problem for harmonic maps on Minkowski space, in Proceed. Inter. Congress of Math. 1994, Birkhäuser, 1126–1132. | MR | Zbl

[48] J. Shatah, W. Strauss, Breathers as homoclinic geometric wave maps, Physics D 99 (1996), 113–133. | MR | Zbl

[49] J. Shatah, M. Struwe, Regularity results for non-linear wave equations, Ann. of Math. 138 (1993) 503–518. | MR | Zbl

[50] J. Shatah, M. Struwe, Geometric Wave Equations, Courant Lecture Notes in Mathematics 2 (1998) | MR | Zbl

[51] J. Shatah, M. Struwe, The Cauchy problem for wave maps, IMRN 11 (2002) 555–571. | MR | Zbl

[52] J. Shatah, A. Tavildar-Zadeh, Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds., Comm. Pure Appl. Math. 45 (1992), 947–971. | MR | Zbl

[53] J. Shatah, A. Tavildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719 – 753. | MR | Zbl

[54] J. Shatah, A. Tavildar-Zadeh, On the stability of stationary wave maps, Comm. Math. Phys., 185 (1996), 231 – 256. | MR | Zbl

[55] T. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math., 42 (1989),1–13. | MR | Zbl

[56] C. D. Sogge, Lectures on Nonlinear Wave Equations, Monographs in Analysis II, International Press, 1995. | MR | Zbl

[57] E. M. Stein, Harmonic Analysis, Princeton University Press, 1993. | MR | Zbl

[58] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), no. 4, 558–581. | MR | Zbl

[59] M. Struwe, Wave Maps, in Nonlinear Partial Differential Equations in Geometry and Physics, Prog. in Nonlin. Diff. Eq. and their Applic., 29, (1997), Birkhäuser, 113–150. | MR | Zbl

[60] M. Struwe, Radially symmetric wave maps from the (1+2)-dimensional Minkowski space to a sphere, Math Z. 242 (2002), 407–414. | MR | Zbl

[61] M. Struwe, Radially symmetric wave maps from (1+2)-dimensional Minkowski space to general targets, Calc. Var. 16 (2003), 431–437. | MR | Zbl

[62] M. Struwe, Equivariant wave maps in two dimensions, preprint.

[63] T. Tao, Ill-posedness for one-dimensional wave maps at the critical regularity, Amer. J. Math. 122 (2000), 451–463. | MR | Zbl

[64] T. Tao, Global regularity of wave maps I. Small critical Sobolev norm in high dimension, IMRN 7 (2001), 299-328. | MR | Zbl

[65] T. Tao, Global regularity of wave maps II. Small energy in two dimensions, submitted, Comm. Math. Phys. | MR | Zbl

[66] D. Tataru, Local and global results for wave maps I, Comm. PDE 23 (1998), 1781–1793. | MR | Zbl

[67] D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. | MR | Zbl

[68] D. Tataru, Rough solutions for the wave maps equation, preprint. | Zbl

[69] D. Tataru, The wave maps equation, preprint. | MR | Zbl