Selfsimilar expanders of the harmonic map flow
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 743-773.

On étudie lʼexistence, lʼunicité et la stabilité de solutions auto-similaires issues dʼune singularité, pour le flot gradient des applications harmoniques, dans le cadre équivariant. On montre lʼexistence de telles solutions auto-similaires, et comment leurs propriétés dʼunicité et de stabilité sont étroitement reliées à la minimisation ou non de lʼénergie de Dirichlet par lʼapplication équateur.

We study the existence, uniqueness, and stability of self-similar expanders of the harmonic map heat flow in equivariant settings. We show that there exist selfsimilar solutions to any admissible initial data and that their uniqueness and stability properties are essentially determined by the energy-minimising properties of the so-called equator maps.

@article{AIHPC_2011__28_5_743_0,
author = {Germain, Pierre and Rupflin, Melanie},
title = {Selfsimilar expanders of the harmonic map flow},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {743--773},
publisher = {Elsevier},
volume = {28},
number = {5},
year = {2011},
doi = {10.1016/j.anihpc.2011.06.004},
zbl = {1246.35059},
mrnumber = {2838400},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2011.06.004/}
}
Germain, Pierre; Rupflin, Melanie. Selfsimilar expanders of the harmonic map flow. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 5, pp. 743-773. doi : 10.1016/j.anihpc.2011.06.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2011.06.004/

[1] M. Bertsch, R. Dal Passo, R. Van Der Hout, Nonuniqueness for the heat flow of harmonic maps on the disk, Arch. Rat. Mech. Anal. 161 no. 2 (2002), 93-112 | MR 1870959 | Zbl 1006.35050

[2] P. Biernat, P. Bizon, private communication.

[3] H. Brezis, J.M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986), 649-705 | MR 868739 | Zbl 0608.58016

[4] T. Cazenave, J. Shatah, 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 no. 3 (1998), 315-349 | EuDML 76787 | Numdam | MR 1622539 | Zbl 0918.58074

[5] Y. Chen, The weak solutions to the evolution problems of harmonic maps, Math. Z. 201 no. 1 (1989), 69-74 | EuDML 174035 | MR 990189 | Zbl 0685.58015

[6] Y. Chen, M. Struwe, Existence and partial regularity results for the heat flow for harmonic maps, J. Differential Geometry 201 no. 1 (1989), 83-103 | EuDML 174037 | MR 990191 | Zbl 0652.58024

[7] E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, International Series in Pure and Applied Mathematics, McGraw–Hill (1955) | MR 69338 | Zbl 0042.32602

[8] J.-M. Coron, Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. H. Poincaré, Analyse Non Linéaire 7 no. 4 (1990), 335-344 | EuDML 78227 | Numdam | MR 1067779 | Zbl 0707.58017

[9] J. Eells, A. Ratto, Harmonic Maps and Minimal Immersions with Symmetries, Annals of Mathematics Studies, Princeton University Press (1993) | MR 1242555 | Zbl 0783.58003

[10] M. Escobedo, O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal. 11 no. 10 (1987), 1103-1133 | MR 913672 | Zbl 0639.35038

[11] H. Fan, Existence of the self-similar solutions in the heat flow of harmonic maps, Sci. China, Ser. A 42 (1999), 113-132 | MR 1694169 | Zbl 0926.35021

[12] A. Freire, Uniqueness of the harmonic map flow in two dimensions, Calc. Var. 3 no. 1 (1995), 95-105 | MR 1384838 | Zbl 0814.35057

[13] V. Galaktionov, J. Vazquez, Continuation of blowup solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 no. 1 (1997), 1-67 | MR 1423231 | Zbl 0874.35057

[14] A. Gastel, Regularity theory for minimizing equivariant (p-)harmonic mappings, Calc. Var. 6 no. 4 (1998), 329-367 | MR 1624296 | Zbl 0911.58010

[15] A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom. 6 no. 10 (2006), 595-613 | MR 2267035 | Zbl 1136.58010

[16] P. Germain, On the existence of smooth self-similar blowup profiles for the wave map equation, Comm. Pure Appl. Math. 62 no. 5 (2009), 706-728 | MR 2494812 | Zbl 1179.35033

[17] A. Haraux, F. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 no. 2 (1982), 167-189 | MR 648169 | Zbl 0465.35049

[18] M.-C. Hong, Some new examples for nonuniqueness of the evolution problem of harmonic maps, Comm. Anal. Geom. 6 no. 4 (1998), 809-818 | MR 1664892 | Zbl 0949.58016

[19] T. Ilmanen, Lectures on mean curvature flow and related equations, Lecture Notes ICTP, Trieste, 1995.

[20] H. Kaul, W. Jäger, Rotationally symmetric harmonic maps from a ball into a sphere and the regularity problem for weak solutions of elliptic systems, J. Reine Angew. Math. 343 (1983), 146-161 | EuDML 152552 | MR 705882 | Zbl 0516.35032

[21] H. Koch, T. Lamm, Geometric flows with rough initial data, arXiv:0902.1488v1 (2009) | MR 2916362

[22] F. Lin, C. Wang, On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals, Chinese Ann. Math. Ser. B 31 no. 6 (2010), 921-938 | MR 2745211 | Zbl 1208.35002

[23] F. Lin, C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2008) | MR 2431658

[24] Y. Naito, Non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data, Math. Ann. 329 no. 1 (2004), 161-196 | MR 2052872 | Zbl 1059.35055

[25] Y. Naito, An ode approach to the multiplicity of self-similar solutions for semi-linear heat equations, Proc. Roy. Soc. Edinburgh Sect. A 4 (2006), 807-835 | MR 2250448 | Zbl 1112.35100

[26] L. Peletier, D. Terman, F. Weissler, On the equation $\delta u+\frac{1}{2}x\Delta \nabla u+f\left(u\right)=0$, Arch. Rat. Mech. Anal. 94 no. 1 (1986), 83-99 | MR 831771 | Zbl 0615.35034

[27] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag (1984) | MR 762825 | Zbl 0153.13602

[28] P. Quittner, P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts Basler Lehrbücher, Birkhäuser Verlag, Basel (2007) | MR 2346798 | Zbl 1128.35003

[29] M. Reed, B. Simon, Methods of Modern Mathematical Physics, vol. 4: Analysis of Operators, Academic Press (1972)

[30] J. Rubinstein, P. Sternberg, J. Keller, Reaction-diffusion processes and evolution to harmonic maps, SIAM J. Appl. Math. 49 no. 6 (1995), 1722-1733 | MR 1025956 | Zbl 0702.35128

[31] M. Rupflin, An improved uniqueness result for the harmonic map flow in two dimensions, Calc. Var. 33 no. 3 (2008), 329-341 | MR 2429534 | Zbl 1157.58004

[32] M. Rupflin, Harmonic map flow and variants, PhD thesis, ETH Zurich, 2010.

[33] J. Shatah, Weak solutions and development of singularities of the $\mathrm{su}\left(2\right)$ σ-model, Comm. Pure Appl. Math. 41 no. 4 (1988), 459-469 | MR 933231 | Zbl 0686.35081

[34] R. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs vol. 49, American Mathematical Society, Providence, RI (1997) | MR 1422252 | Zbl 0870.35004

[35] P. Souplet, F. Weissler, Regular self-similar solutions of the nonlinear heat equation with initial data above the singular steady state, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 no. 2 (2003), 213-235 | EuDML 78577 | Numdam | MR 1961515 | Zbl 1029.35106

[36] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Helv. 60 (1985), 558-581 | EuDML 140031 | MR 826871 | Zbl 0595.58013

[37] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geometry 28 no. 3 (1988), 485-502 | MR 965226 | Zbl 0631.58004

[38] P. Topping, Reverse bubbling and nonuniqueness in the harmonic map flow, Int. Math. Research Notices 10 (2002), 558-581 | MR 1883901 | Zbl 1003.58014

[39] J.L. Vazquez, E. Zuazua, The hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal. 173 (2000), 103-153 | MR 1760280 | Zbl 0953.35053

[40] C.-Y. Wang, Bubble phenomena of certain palais-smale sequences from surfaces to general targets, Houston J. Math. 22 (1996), 559-590 | MR 1417632 | Zbl 0879.58019

[41] F. Weissler, Rapidly decaying solutions of an ordinary differential equation with applications to semilinear elliptic and parabolic partial differential equations, Arch. Rat. Mech. Anal. 91 no. 3 (1985), 247-266 | MR 806004 | Zbl 0604.34034