# Article

 Title: Parabolic equations with rough data (English) Author: Koch, Herbert Author: Lamm, Tobias Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 140 Issue: 4 Year: 2015 Pages: 457-477 Summary lang: English . Category: math . 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. (English) Keyword: parabolic equation Keyword: rough initial data MSC: 35K59 MSC: 53C44 idZBL: Zbl 06537677 . Date available: 2015-11-17T20:53:15Z Last updated: 2017-01-02 Stable URL: http://hdl.handle.net/10338.dmlcz/144463 . Reference: [1] Angenent, S. B.: Nonlinear analytic semiflows.Proc. R. Soc. Edinb., Sect. A, Math. 115 (1990), 91-107. Zbl 0723.34047, MR 1059647, 10.1017/S0308210500024598 Reference: [2] Angenent, S. B.: Parabolic equations for curves on surfaces. I: Curves with $p$-integrable curvature.Ann. Math. (2) 132 (1990), 451-483. Zbl 0789.58070, MR 1078266 Reference: [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. Zbl 0780.35079, MR 1208420, 10.1017/S095679250000098X Reference: [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 Reference: [5] Cabezas-Rivas, E., Wilking, B.: How to produce a Ricci flow via Cheeger-{G}romoll exhaustion.(to appear) in J. Eur. Math. Soc. Reference: [6] Chen, B.-L.: Strong uniqueness of the Ricci flow.J. Differ. Geom. 82 (2009), 363-382. Zbl 1177.53036, MR 2520796 Reference: [7] Chen, B.-L., Zhu, X.-P.: Uniqueness of the Ricci flow on complete noncompact manifolds.J. Differ. Geom. 74 (2006), 119-154. Zbl 1104.53032, MR 2260930 Reference: [8] Dahlberg, B. E. J., Kenig, C. E.: Non-negative solutions of generalized porous medium equations.Rev. Mat. Iberoam. 2 (1986), 267-305. Zbl 0644.35057, MR 0908054, 10.4171/RMI/34 Reference: [9] Daskalopoulos, P., Hamilton, R.: Regularity of the free boundary for the porous medium equation.J. Am. Math. Soc. 11 (1998), 899-965. Zbl 0910.35145, MR 1623198, 10.1090/S0894-0347-98-00277-X Reference: [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. Zbl 1017.35052, MR 1833393, 10.1215/S0012-7094-01-10824-7 Reference: [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. Zbl 1315.35004, MR 3307161 Reference: [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. Zbl 1083.35074, MR 2126633, 10.1007/s00205-004-0336-3 Reference: [13] DeTurck, D. M.: Deforming metrics in the direction of their Ricci tensors.J. Differ. Geom. 18 (1983), 157-162. Zbl 0517.53044, MR 0697987 Reference: [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. Zbl 1302.35218, MR 3197240, 10.1016/j.jde.2014.03.010 Reference: [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. Zbl 1159.35039, MR 2436450, 10.1016/j.jde.2008.06.005 Reference: [16] Jerison, D., Kenig, C. E.: The inhomogeneous Dirichlet problem in Lipschitz domains.J. Funct. Anal. 130 (1995), 161-219. Zbl 0832.35034, MR 1331981, 10.1006/jfan.1995.1067 Reference: [17] John, D.: On uniqueness of weak solutions for the thin-film equation.J. Differ. Equations 259 (2015), Article ID 7877, 4122-4171. Zbl 1322.35084, MR 3369273, 10.1016/j.jde.2015.05.013 Reference: [18] Kienzler, C.: Flat Fronts and Stability for the Porous Medium Equation.(2014), \hfil arxiv:1403.5811[math.AP]. Reference: [19] Koch, H., Lamm, T.: Geometric flows with rough initial data.Asian J. Math. 16 (2012), 209-235. Zbl 1252.35159, MR 2916362, 10.4310/AJM.2012.v16.n2.a3 Reference: [20] Koch, H., Tataru, D.: Well-posedness for the Navier-Stokes equations.Adv. Math. 157 (2001), 22-35. Zbl 0972.35084, MR 1808843, 10.1006/aima.2000.1937 Reference: [21] Kotschwar, B. L.: An energy approach to the problem of uniqueness for the Ricci flow.Commun. Anal. Geom. 22 (2014), 149-176. Zbl 1303.53056, MR 3194377, 10.4310/CAG.2014.v22.n1.a3 Reference: [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. Zbl 1259.53065, MR 3033963, 10.1112/blms/bds074 Reference: [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. Zbl 1257.35092, MR 2964616, 10.1016/j.aim.2012.07.005 Reference: [24] Nadirashvili, N., Vlăduţ, S.: Nonclassical solutions of fully nonlinear elliptic equations.Geom. Funct. Anal. 17 (2007), 1283-1296. Zbl 1132.35036, MR 2373018, 10.1007/s00039-007-0626-7 Reference: [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 Reference: [26] Shao, Y., Simonett, G.: Continuous maximal regularity on uniformly regular Riemannian manifolds.J. Evol. Equ. 14 (2014), 211-248. Zbl 1295.35161, MR 3169036, 10.1007/s00028-014-0218-6 Reference: [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. Zbl 1239.53085, MR 2876261 Reference: [28] Simon, M.: Deformation of $C^0$ Riemannian metrics in the direction of their Ricci curvature.Commun. Anal. Geom. 10 (2002), 1033-1074. Zbl 1034.58008, MR 1957662, 10.4310/CAG.2002.v10.n5.a7 Reference: [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. Zbl 0526.73026, MR 0713118, 10.1007/BF00251549 Reference: [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. Zbl 0164.12502, MR 0211083 Reference: [31] Šverák, V.: Rank-one convexity does not imply quasiconvexity.Proc. R. Soc. Edinb., Sect. A, Math. 120 (1992), 185-189. Zbl 0777.49015, MR 1149994, 10.1017/S0308210500015080 Reference: [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. Zbl 1285.35085, MR 2781584, 10.1007/s00205-010-0343-5 Reference: [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. Zbl 1071.35050, MR 2012810, 10.1002/cpa.10117 Reference: [34] Wang, M.-T.: The mean curvature flow smoothes Lipschitz submanifolds.Commun. Anal. Geom. 12 (2004), 581-599. Zbl 1059.53053, MR 2128604, 10.4310/CAG.2004.v12.n3.a4 Reference: [35] Whitney, H.: The imbedding of manifolds in families of analytic manifolds.Ann. Math. (2) 37 (1936), 865-878. Zbl 0015.18002, MR 1503315, 10.2307/1968624 .

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