Previous |  Up |  Next

Article

Keywords:
parabolic equation; rough initial data
Summary:
We survey recent work on local well-posedness results for parabolic equations and systems with rough initial data. The design of the function spaces is guided by tools and constructions from harmonic analysis, like maximal functions, square functions and Carleson measures. We construct solutions under virtually optimal scale invariant conditions on the initial data. Applications include BMO initial data for the harmonic map heat flow and the Ricci-DeTurck flow for initial metrics with small local oscillation. The approach is sufficiently flexible to apply to boundary value problems, quasilinear and fully nonlinear equations.
References:
[1] Angenent, S. B.: Nonlinear analytic semiflows. Proc. R. Soc. Edinb., Sect. A, Math. 115 (1990), 91-107. DOI 10.1017/S0308210500024598 | MR 1059647 | Zbl 0723.34047
[2] Angenent, S. B.: Parabolic equations for curves on surfaces. I: Curves with $p$-integrable curvature. Ann. Math. (2) 132 (1990), 451-483. MR 1078266 | Zbl 0789.58070
[3] Aronson, D. G., Graveleau, J.: A selfsimilar solution to the focusing problem for the porous medium equation. Eur. J. Appl. Math. 4 (1993), 65-81. DOI 10.1017/S095679250000098X | MR 1208420 | Zbl 0780.35079
[4] Auscher, P., Hofmann, S., Lacey, M., McIntosh, A., Tchamitchian, P.: The solution of the Kato square root problem for second order elliptic operators on $\mathbb R^n$. Ann. Math. (2) 156 (2002), 633-654. MR 1933726
[5] Cabezas-Rivas, E., Wilking, B.: How to produce a Ricci flow via Cheeger-{G}romoll exhaustion. (to appear) in J. Eur. Math. Soc.
[6] Chen, B.-L.: Strong uniqueness of the Ricci flow. J. Differ. Geom. 82 (2009), 363-382. MR 2520796 | Zbl 1177.53036
[7] Chen, B.-L., Zhu, X.-P.: Uniqueness of the Ricci flow on complete noncompact manifolds. J. Differ. Geom. 74 (2006), 119-154. MR 2260930 | Zbl 1104.53032
[8] Dahlberg, B. E. J., Kenig, C. E.: Non-negative solutions of generalized porous medium equations. Rev. Mat. Iberoam. 2 (1986), 267-305. DOI 10.4171/RMI/34 | MR 0908054 | Zbl 0644.35057
[9] Daskalopoulos, P., Hamilton, R.: Regularity of the free boundary for the porous medium equation. J. Am. Math. Soc. 11 (1998), 899-965. DOI 10.1090/S0894-0347-98-00277-X | MR 1623198 | Zbl 0910.35145
[10] Daskalopoulos, P., Hamilton, R., Lee, K.: All time $C^\infty$-regularity of the interface in degenerate diffusion: A geometric approach. Duke Math. J. 108 (2001), 295-327. DOI 10.1215/S0012-7094-01-10824-7 | MR 1833393 | Zbl 1017.35052
[11] Denzler, J., Koch, H., McCann, R. J.: Higher-order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach. Mem. Am. Math. Soc. 234 (2015), no. 1101, 81 pages. MR 3307161 | Zbl 1315.35004
[12] Denzler, J., McCann, R. J.: Fast diffusion to self-similarity: Complete spectrum, long-time asymptotics, and numerology. Arch. Ration. Mech. Anal. 175 (2005), 301-342. DOI 10.1007/s00205-004-0336-3 | MR 2126633 | Zbl 1083.35074
[13] DeTurck, D. M.: Deforming metrics in the direction of their Ricci tensors. J. Differ. Geom. 18 (1983), 157-162. MR 0697987 | Zbl 0517.53044
[14] Giacomelli, L., Gnann, M. V., Knüpfer, H., Otto, F.: Well-posedness for the Navier-slip thin-film equation in the case of complete wetting. J. Differ. Equations 257 (2014), 15-81. DOI 10.1016/j.jde.2014.03.010 | MR 3197240 | Zbl 1302.35218
[15] Giacomelli, L., Knüpfer, H., Otto, F.: Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equations 245 (2008), 1454-1506. DOI 10.1016/j.jde.2008.06.005 | MR 2436450 | Zbl 1159.35039
[16] Jerison, D., Kenig, C. E.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130 (1995), 161-219. DOI 10.1006/jfan.1995.1067 | MR 1331981 | Zbl 0832.35034
[17] John, D.: On uniqueness of weak solutions for the thin-film equation. J. Differ. Equations 259 (2015), Article ID 7877, 4122-4171. DOI 10.1016/j.jde.2015.05.013 | MR 3369273 | Zbl 1322.35084
[18] Kienzler, C.: Flat Fronts and Stability for the Porous Medium Equation. (2014), \hfil arxiv:1403.5811[math.AP].
[19] Koch, H., Lamm, T.: Geometric flows with rough initial data. Asian J. Math. 16 (2012), 209-235. DOI 10.4310/AJM.2012.v16.n2.a3 | MR 2916362 | Zbl 1252.35159
[20] Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157 (2001), 22-35. DOI 10.1006/aima.2000.1937 | MR 1808843 | Zbl 0972.35084
[21] Kotschwar, B. L.: An energy approach to the problem of uniqueness for the Ricci flow. Commun. Anal. Geom. 22 (2014), 149-176. DOI 10.4310/CAG.2014.v22.n1.a3 | MR 3194377 | Zbl 1303.53056
[22] Kotschwar, B. L.: A local version of Bando's theorem on the real-analyticity of solutions to the Ricci flow. Bull. Lond. Math. Soc. 45 (2013), 153-158. DOI 10.1112/blms/bds074 | MR 3033963 | Zbl 1259.53065
[23] Nadirashvili, N., Tkachev, V., Vlăduţ, S.: A non-classical solution to a Hessian equation from Cartan isoparametric cubic. Adv. Math. 231 (2012), 1589-1597. DOI 10.1016/j.aim.2012.07.005 | MR 2964616 | Zbl 1257.35092
[24] Nadirashvili, N., Vlăduţ, S.: Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17 (2007), 1283-1296. DOI 10.1007/s00039-007-0626-7 | MR 2373018 | Zbl 1132.35036
[25] Shao, Y.: A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows. arXiv:1309.2043 (2013), 36 pages. MR 3311893
[26] Shao, Y., Simonett, G.: Continuous maximal regularity on uniformly regular Riemannian manifolds. J. Evol. Equ. 14 (2014), 211-248. DOI 10.1007/s00028-014-0218-6 | MR 3169036 | Zbl 1295.35161
[27] Simon, M.: Ricci flow of non-collapsed three manifolds whose Ricci curvature is bounded from below. J. Reine Angew. Math. 662 (2012), 59-94. MR 2876261 | Zbl 1239.53085
[28] Simon, M.: Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature. Commun. Anal. Geom. 10 (2002), 1033-1074. DOI 10.4310/CAG.2002.v10.n5.a7 | MR 1957662 | Zbl 1034.58008
[29] Simpson, H. C., Spector, S. J.: On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Ration. Mech. Anal. 84 (1983), 55-68. DOI 10.1007/BF00251549 | MR 0713118 | Zbl 0526.73026
[30] Solonnikov, V. A.: On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov. 83 (1965), 3-163. MR 0211083 | Zbl 0164.12502
[31] Šverák, V.: Rank-one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb., Sect. A, Math. 120 (1992), 185-189. DOI 10.1017/S0308210500015080 | MR 1149994 | Zbl 0777.49015
[32] Wang, C.: Well-posedness for the heat flow of harmonic maps and the liquid crystal flow with rough initial data. Arch. Ration. Mech. Anal. 200 (2011), 1-19. DOI 10.1007/s00205-010-0343-5 | MR 2781584 | Zbl 1285.35085
[33] Wang, M.-T.: The Dirichlet problem for the minimal surface system in arbitrary dimensions and codimensions. Commun. Pure Appl. Math. 57 (2004), 267-281. DOI 10.1002/cpa.10117 | MR 2012810 | Zbl 1071.35050
[34] Wang, M.-T.: The mean curvature flow smoothes Lipschitz submanifolds. Commun. Anal. Geom. 12 (2004), 581-599. DOI 10.4310/CAG.2004.v12.n3.a4 | MR 2128604 | Zbl 1059.53053
[35] Whitney, H.: The imbedding of manifolds in families of analytic manifolds. Ann. Math. (2) 37 (1936), 865-878. DOI 10.2307/1968624 | MR 1503315 | Zbl 0015.18002
Partner of
EuDML logo