The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in [
Citation: Jerome D. Wettstein. Half-harmonic gradient flow: aspects of a non-local geometric PDE[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023058
The goal of this paper is to discuss some of the results in the author's previous papers and expand upon the work there by proving two new results: a global weak existence result as well as a first bubbling analysis for the half-harmonic gradient flow in finite time. In addition, an alternative local existence proof to the one provided in [
[1] | A. Audrito, On the existence and Hölder regularity of solutions to some nonlinear Cauchy-Neumann problems, arXiv: 2107.03308. |
[2] | L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Commun. Part. Diff. Eq., 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306 doi: 10.1080/03605300600987306 |
[3] | K.-C. Chang, W. Y. Ding, R. Ye, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom., 36 (1992), 507–515. https://doi.org/10.4310/jdg/1214448751 doi: 10.4310/jdg/1214448751 |
[4] | R. Coifman, P. Lions, Y. Meyer, S. Semmes, Compensated compactness and Hardy spaces, Journal de mathématiques pures et appliquées, 72 (1993), 247–286. |
[5] | F. Da Lio, Compactness and bubbles analysis for half-harmonic maps into spheres, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 201–224. https://doi.org/10.1016/j.anihpc.2013.11.003 doi: 10.1016/j.anihpc.2013.11.003 |
[6] | F. Da Lio, Fractional harmonic maps into manifolds in odd dimensions $> 1$, Calc. Var., 48 (2013), 421–445. https://doi.org/10.1007/s00526-012-0556-6 doi: 10.1007/s00526-012-0556-6 |
[7] | F. Da Lio, P. Laurain, T. Rivière, A Pohozaev-type formula and quantization of horizontal half-harmonic maps, arXiv: 1607.05504. |
[8] | F. Da Lio, Fractional harmonic maps, In: Recent developments in nonlocal theory, Warsaw, Poland: De Gruyter, 2018, 52–80. https://doi.org/10.1515/9783110571561-004 |
[9] | F. Da Lio, A. Pigati, Free boundary minimal surfaces: a nonlocal approach, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XX (2020), 437–489. https://doi.org/10.2422/2036-2145.201801_008 doi: 10.2422/2036-2145.201801_008 |
[10] | F. Da Lio, T. Rivière, 3-Commutator estimates and the regularity of $1/2$-harmonic maps into spheres, Anal. PDE, 4 (2011), 149–190. https://doi.org/10.2140/apde.2011.4.149 doi: 10.2140/apde.2011.4.149 |
[11] | F. Da Lio, T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., 277 (2011), 1300–1348. https://doi.org/10.1016/j.aim.2011.03.011 doi: 10.1016/j.aim.2011.03.011 |
[12] | F. Da Lio, A. Schikorra, n/p-Harmonic maps: regularity for the sphere case, Adv. Calc. Var., 7 (2014), 1–26. https://doi.org/10.1515/acv-2012-0107 doi: 10.1515/acv-2012-0107 |
[13] | F. Da Lio, A. Schikorra, On regularity theory for n/p-harmonic maps into manifolds, Nonlinear Anal., 165 (2017), 182–197. https://doi.org/10.1016/j.na.2017.10.001 doi: 10.1016/j.na.2017.10.001 |
[14] | J. Davila, M. Del Pino, J. Wei, Singularity formation for the two-dimensional harmonic map flow in $S^2$, Invent. Math., 219 (2020), 345–466. https://doi.org/10.1007/s00222-019-00908-y doi: 10.1007/s00222-019-00908-y |
[15] | J. Eells, J. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math., 86 (1964), 109–160. |
[16] | A. Freire, Uniqueness for the harmonic map flow from surfaces into general targets, Commentarii Mathematici Helvetici, 70 (1995), 310–338. https://doi.org/10.1007/BF02566010 doi: 10.1007/BF02566010 |
[17] | L. Grafakos, Modern Fourier analysis, 2 Eds., New York: Springer, 2009. https://doi.org/10.1007/978-0-387-09434-2 |
[18] | M. Grüter, Regularity of weak H-surfaces, J. Reine Angew. Math., 1981 (1981), 1–15. https://doi.org/10.1515/crll.1981.329.1 doi: 10.1515/crll.1981.329.1 |
[19] | F. Hélein, Régularité des applications faiblement harmoniques entre une surface et une varitée riemannienne, C. R. Acad. Sci. Paris Sr. I Math., 311 (1990), 591–596. |
[20] | F. Hélein, Harmonic maps, conservation laws and moving frames, 2 Eds., Cambridge: Cambridge University Press, 2002. https://doi.org/10.1017/CBO9780511543036 |
[21] | A. Hyder, A. Segatti, Y. Sire, C. Wang, Partial regularity of the heat flow of half-harmonic maps and applications to harmonic maps with free boundary, Commun Part. Diff. Eq., 47 (2022), 1845–1882. https://doi.org/10.1080/03605302.2022.2091453 doi: 10.1080/03605302.2022.2091453 |
[22] | F. John, Partial differential equations, 3 Eds., New York: Springer, 1978. https://doi.org/10.1007/978-1-4684-0059-5 |
[23] | J. Jost, Geometry and physics, Heidelberg: Springer, 2009. https://doi.org/10.1007/978-3-642-00541-1 |
[24] | O. A. Ladyzhenskaya, Solutions "in the large" of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Commun. Pure Appl. Math., 7 (1959), 427–433. https://doi.org/10.1002/cpa.3160120303 doi: 10.1002/cpa.3160120303 |
[25] | K. Mazowiecka, A. Schikorra, Fractional div-curl quantities and applications to nonlocal geometric equation, J. Funct. Anal., 275 (2018), 1–44. https://doi.org/10.1016/j.jfa.2018.03.016 doi: 10.1016/j.jfa.2018.03.016 |
[26] | V. Millot, Y. Sire, On a fractional Ginzburg-Landau equation and $1/2$-harmonic maps into spheres, Arch. Rational Mech. Anal., 215 (2015), 125–210. https://doi.org/10.1007/s00205-014-0776-3 doi: 10.1007/s00205-014-0776-3 |
[27] | C. Morrey, The problem of plateau on a Riemannian manifold, Ann. Math., 49 (1948), 807–851. |
[28] | E. Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004 |
[29] | M. Prats, Measuring Triebel-Lizorkin fractional smoothness on domains in terms of first-order differences, J. London Math. Soc., 100 (2019), 692–716. https://doi.org/10.1112/jlms.12225 doi: 10.1112/jlms.12225 |
[30] | M. Prats, E. Saksman, A $T(1)$ theorem for fractional Sobolev spaces on domains, J. Geom. Anal., 27 (2017), 2490–2538. https://doi.org/10.1007/s12220-017-9770-y doi: 10.1007/s12220-017-9770-y |
[31] | T. Rivière, Le flot des applications faiblement harmoniques en dimension deux, PhD thesis, 1993. |
[32] | T. Rivière, Conservation laws for conformally invariant variational problems, Invent. Math., 168 (2007), 1–22. https://doi.org/10.1007/s00222-006-0023-0 doi: 10.1007/s00222-006-0023-0 |
[33] | T. Rivière, Conformally invariant variational problems, arXiv: 1206.2116. |
[34] | J. Sacks, K. Uhlenbeck, The existence of minimal immersions of $2$-spheres, Ann. Math., 113 (1981), 1–24. https://doi.org/10.2307/1971131 doi: 10.2307/1971131 |
[35] | A. Schikorra, Regularity of n/2-harmonic maps into the sphere, J. Differ. Equations, 252 (2012), 1862–1911. https://doi.org/10.1016/j.jde.2011.08.021 doi: 10.1016/j.jde.2011.08.021 |
[36] | A. Schikorra, Y. Sire, C. Wang, Weak solutions of geometric flows associated to integro-differential harmonic maps, Manuscripta Math., 153 (2017), 389–402. https://doi.org/10.1007/s00229-016-0899-y doi: 10.1007/s00229-016-0899-y |
[37] | R. Schoen, S. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds with non-negative Ricci curvature, Commentarii Mathematici Helvetici, 51 (1976), 333–341. https://doi.org/10.1007/BF02568161 doi: 10.1007/BF02568161 |
[38] | H. Schmeisser, H. Triebel, Topics in Fourier analysis and function spaces, Chichester: J. Wiley, 1987. |
[39] | J. Shatah, Weak solutions and development of singularities of the $SU(2)$ $\sigma$-model, Commun Pure Appl. Math., 41 (1988), 459–469. https://doi.org/10.1002/cpa.3160410405 doi: 10.1002/cpa.3160410405 |
[40] | Y. Sire, J. Wei, Y. Zheng, Infinite time blow-up for half-harmonic map flow from $ \mathbb{R}$ into $S^1$, arXiv: 1711.05387. |
[41] | M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Commentarii Mathematici Helvetici, 60 (1985), 558–581. https://doi.org/10.1007/BF02567432 doi: 10.1007/BF02567432 |
[42] | M. Struwe, On the evolution of harmonic maps in higher dimension, J. Differential Geom., 28 (1988), 485–502. https://doi.org/10.4310/jdg/1214442475 doi: 10.4310/jdg/1214442475 |
[43] | M. Struwe, Plateau flow or the heat flow for half-harmonic maps, arXiv: 2202.02083. |
[44] | P. Topping, Reverse bubbling and nonuniqueness in the harmonic map flow, Int. Math. Res. Notices, 10 (2002), 505–520. https://doi.org/10.1155/S1073792802105083 doi: 10.1155/S1073792802105083 |
[45] | K. Uhlenbeck, Connections with $L^p$ bounds on curvature, Commun. Math. Phys., 83 (1982), 31–42. https://doi.org/10.1007/BF01947069 doi: 10.1007/BF01947069 |
[46] | H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl., 26 (1969), 318–344. https://doi.org/10.1016/0022-247X(69)90156-5 doi: 10.1016/0022-247X(69)90156-5 |
[47] | J. Wettstein, Uniqueness and regularity of the fractional harmonic gradient flow in $S^{n-1}$, Nonlinear Anal., 214 (2022), 112592. https://doi.org/10.1016/j.na.2021.112592 doi: 10.1016/j.na.2021.112592 |
[48] | J. Wettstein, Existence, uniqueness and regularity of the fractional harmonic gradient flow in general target manifolds, arXiv: 2109.11458. |