# American Institute of Mathematical Sciences

June  2016, 5(2): 303-335. doi: 10.3934/eect.2016006

## Continuous maximal regularity on singular manifolds and its applications

 1 Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907, United States

Received  December 2015 Revised  March 2016 Published  June 2016

In this article, we set up the continuous maximal regularity theory for a class of linear differential operators on manifolds with singularities. These operators exhibit degenerate or singular behaviors while approaching the singular ends. Particular examples of such operators include differential operators defined on domains, which degenerate fast enough toward the boundary. Applications of the theory established herein are shown to the Yamabe flow, the porous medium equation, the parabolic $p$-Laplacian equation and the thin film equation. Some comments about the boundary blow-up problem, and waiting time phenomenon for singular or degenerate parabolic equations can also be found in this paper.
Citation: Yuanzhen Shao. Continuous maximal regularity on singular manifolds and its applications. Evolution Equations & Control Theory, 2016, 5 (2) : 303-335. doi: 10.3934/eect.2016006
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems: Volume I.,, Abstract linear theory. Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] H. Amann, Elliptic operators with infinite-dimensional state spaces,, J. Evol. Equ., 1 (2001), 143. doi: 10.1007/PL00001367. Google Scholar [3] H. Amann, Function spaces on singular manifolds,, Math. Nachr., 286 (2013), 436. doi: 10.1002/mana.201100157. Google Scholar [4] H. Amann, Anisotropic function spaces on singular manifolds,, , (). doi: 10.1002/mana.201100157. Google Scholar [5] H. Amann, Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems,, Recent Developments of Mathematical Fluid Mechanics, (2016). doi: 10.1007/978-3-0348-0939-9_4. Google Scholar [6] H. Amann, Uniformly Regular and Singular Riemannian Manifolds,, Elliptic and parabolic equations, 119 (2015), 1. doi: 10.1007/978-3-319-12547-3_1. Google Scholar [7] H. Amann, Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and Hölder Spaces on Uniformly Regular Riemannian Manifolds,, , (). Google Scholar [8] S. B. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598. Google Scholar [9] E. Bahuaud and B. Vertman, Yamabe flow on manifolds with edges,, Math. Nachr., 287 (2014), 127. doi: 10.1002/mana.201200210. Google Scholar [10] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differential Equations, 83 (1990), 179. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [11] M. Boutat, S. Hilout, J.-E. Rakotoson and J.-M. Rakotoson, A generalized thin-film equation in multidimensional space,, Nonlinear Anal., 69 (2008), 1268. doi: 10.1016/j.na.2007.06.028. Google Scholar [12] F. E. Browder, Analyticity and partial differential equations I,, Amer. J. Math., 84 (1962), 666. doi: 10.2307/2372872. Google Scholar [13] D. Chang, N. Habal and B.-W. Schulze, Quantisation on a manifold with singular edge,, J. Pseudo-Differ. Oper. Appl., 4 (2013), 317. doi: 10.1007/s11868-013-0077-x. Google Scholar [14] J. Chang and J. Lee, Harnack-type inequalities for the porous medium equation on a manifold with non-negative Ricci curvature,, Internat. J. Math., 23 (2012). doi: 10.1142/S0129167X11007525. Google Scholar [15] P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations,, J. Evol. Equ., 1 (2001), 39. doi: 10.1007/PL00001364. Google Scholar [16] R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations,, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437. Google Scholar [17] G. Da Prato and P. Grisvard, Equations d'évolution abstraites non linéaires de type parabolique,, Ann. Mat. Pura Appl., 120 (1979), 329. doi: 10.1007/BF02411952. Google Scholar [18] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899. doi: 10.1090/S0894-0347-98-00277-X. Google Scholar [19] S. A. J. Dekkers, A comparison theorem for solutions of degenerate parabolic equations on manifolds,, Proc. Roy. Soc. Edinburgh Sect., 138 (2008), 755. doi: 10.1017/S0308210505000880. Google Scholar [20] J. I. Diáz, T. Nagai and S. I. Shmarëv, On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics,, Japan J. Indust. Appl. Math., 13 (1996), 385. doi: 10.1007/BF03167255. Google Scholar [21] E. Di Benedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar [22] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics. Springer, (2012). doi: 10.1007/978-1-4614-1584-8. Google Scholar [23] M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds,, Math. Nachr., 289 (2016), 232. doi: 10.1002/mana.201400354. Google Scholar [24] S. Fornaro, G. Metafune and D. Pallara, Analytic semigroups generated in Lp by elliptic operators with high order degeneracy at the boundary,, Note Mat., 31 (2011), 103. Google Scholar [25] L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems,, Interfaces Free Bound., 8 (2006), 111. doi: 10.4171/IFB/137. Google Scholar [26] G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon,, Ann. Inst. H. Poincaré Anal., 21 (2004), 255. doi: 10.1016/j.anihpc.2003.02.002. Google Scholar [27] J. R. King, Two generalisations of the thin film equation,, Math. Comput. Modelling, 34 (2001), 737. doi: 10.1016/S0895-7177(01)00095-4. Google Scholar [28] A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows,, Indiana Univ. Math. J., 57 (2008), 2137. doi: 10.1512/iumj.2008.57.3349. Google Scholar [29] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar [30] L. Ma, L. Cheng and A. Zhu, Extending Yamabe flow on complete Riemannian manifolds,, Bull. Sci. Math., 136 (2012), 882. doi: 10.1016/j.bulsci.2012.06.004. Google Scholar [31] R. B. Melrose, Transformation of boundary problems,, Acta Math., 147 (1981), 149. doi: 10.1007/BF02392873. Google Scholar [32] R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem,, Research Notes in Mathematics, (1993). Google Scholar [33] F. Otto and M. Westdickenberg Michael, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227. doi: 10.1137/050622420. Google Scholar [34] J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbbR^n$,, Discrete Contin. Dyn. Syst., 33 (2013), 5407. doi: 10.3934/dcds.2013.33.5407. Google Scholar [35] N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities,, , (). Google Scholar [36] B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics,, Mathematical Topics, (1994). Google Scholar [37] B.-W. Schulze, Boundary Value Problems and Edge Pseudo-Differential Operators,, Microlocal analysis and spectral theory (Lucca, (1996), 165. Google Scholar [38] Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows,, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45. doi: 10.1007/s00030-014-0275-0. Google Scholar [39] Y. Shao, Singular parabolic equations of second order on manifolds with singularities,, J. Differential Equations, 260 (2016), 1747. doi: 10.1016/j.jde.2015.09.053. Google Scholar [40] Y. Shao, The Yamabe flow on incomplete manifolds,, Submitted. , (). Google Scholar [41] Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds,, J. Evol. Equ., 1 (2014), 211. doi: 10.1007/s00028-014-0218-6. Google Scholar [42] A. E. Shishkov, Waiting time of propagation and the backward motion of interfaces in thin-film flow theory,, Discrete Contin. Dyn. Syst., (2007), 938. Google Scholar [43] H. Triebel, Theory of Function Spaces I,, Birkhäuser Verlag, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar [44] J. L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation,, Shape optimization and free boundaries (Montreal, (1990), 347. Google Scholar [45] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, (2007). Google Scholar [46] V. Vespri, Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems,, Ann. Mat. Pura Appl., 155 (1989), 353. doi: 10.1007/BF01765950. Google Scholar [47] X. Xu, Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems,, J. Differential Equations, 252 (2012), 1403. doi: 10.1016/j.jde.2011.08.004. Google Scholar [48] X. Zhu, Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds,, J. Math. Anal. Appl., 402 (2013), 201. doi: 10.1016/j.jmaa.2013.01.018. Google Scholar

show all references

##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems: Volume I.,, Abstract linear theory. Monographs in Mathematics, (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar [2] H. Amann, Elliptic operators with infinite-dimensional state spaces,, J. Evol. Equ., 1 (2001), 143. doi: 10.1007/PL00001367. Google Scholar [3] H. Amann, Function spaces on singular manifolds,, Math. Nachr., 286 (2013), 436. doi: 10.1002/mana.201100157. Google Scholar [4] H. Amann, Anisotropic function spaces on singular manifolds,, , (). doi: 10.1002/mana.201100157. Google Scholar [5] H. Amann, Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems,, Recent Developments of Mathematical Fluid Mechanics, (2016). doi: 10.1007/978-3-0348-0939-9_4. Google Scholar [6] H. Amann, Uniformly Regular and Singular Riemannian Manifolds,, Elliptic and parabolic equations, 119 (2015), 1. doi: 10.1007/978-3-319-12547-3_1. Google Scholar [7] H. Amann, Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and Hölder Spaces on Uniformly Regular Riemannian Manifolds,, , (). Google Scholar [8] S. B. Angenent, Nonlinear analytic semiflows,, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91. doi: 10.1017/S0308210500024598. Google Scholar [9] E. Bahuaud and B. Vertman, Yamabe flow on manifolds with edges,, Math. Nachr., 287 (2014), 127. doi: 10.1002/mana.201200210. Google Scholar [10] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differential Equations, 83 (1990), 179. doi: 10.1016/0022-0396(90)90074-Y. Google Scholar [11] M. Boutat, S. Hilout, J.-E. Rakotoson and J.-M. Rakotoson, A generalized thin-film equation in multidimensional space,, Nonlinear Anal., 69 (2008), 1268. doi: 10.1016/j.na.2007.06.028. Google Scholar [12] F. E. Browder, Analyticity and partial differential equations I,, Amer. J. Math., 84 (1962), 666. doi: 10.2307/2372872. Google Scholar [13] D. Chang, N. Habal and B.-W. Schulze, Quantisation on a manifold with singular edge,, J. Pseudo-Differ. Oper. Appl., 4 (2013), 317. doi: 10.1007/s11868-013-0077-x. Google Scholar [14] J. Chang and J. Lee, Harnack-type inequalities for the porous medium equation on a manifold with non-negative Ricci curvature,, Internat. J. Math., 23 (2012). doi: 10.1142/S0129167X11007525. Google Scholar [15] P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations,, J. Evol. Equ., 1 (2001), 39. doi: 10.1007/PL00001364. Google Scholar [16] R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations,, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437. Google Scholar [17] G. Da Prato and P. Grisvard, Equations d'évolution abstraites non linéaires de type parabolique,, Ann. Mat. Pura Appl., 120 (1979), 329. doi: 10.1007/BF02411952. Google Scholar [18] P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation,, J. Amer. Math. Soc., 11 (1998), 899. doi: 10.1090/S0894-0347-98-00277-X. Google Scholar [19] S. A. J. Dekkers, A comparison theorem for solutions of degenerate parabolic equations on manifolds,, Proc. Roy. Soc. Edinburgh Sect., 138 (2008), 755. doi: 10.1017/S0308210505000880. Google Scholar [20] J. I. Diáz, T. Nagai and S. I. Shmarëv, On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics,, Japan J. Indust. Appl. Math., 13 (1996), 385. doi: 10.1007/BF03167255. Google Scholar [21] E. Di Benedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar [22] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics. Springer, (2012). doi: 10.1007/978-1-4614-1584-8. Google Scholar [23] M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds,, Math. Nachr., 289 (2016), 232. doi: 10.1002/mana.201400354. Google Scholar [24] S. Fornaro, G. Metafune and D. Pallara, Analytic semigroups generated in Lp by elliptic operators with high order degeneracy at the boundary,, Note Mat., 31 (2011), 103. Google Scholar [25] L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems,, Interfaces Free Bound., 8 (2006), 111. doi: 10.4171/IFB/137. Google Scholar [26] G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon,, Ann. Inst. H. Poincaré Anal., 21 (2004), 255. doi: 10.1016/j.anihpc.2003.02.002. Google Scholar [27] J. R. King, Two generalisations of the thin film equation,, Math. Comput. Modelling, 34 (2001), 737. doi: 10.1016/S0895-7177(01)00095-4. Google Scholar [28] A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows,, Indiana Univ. Math. J., 57 (2008), 2137. doi: 10.1512/iumj.2008.57.3349. Google Scholar [29] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser Verlag, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar [30] L. Ma, L. Cheng and A. Zhu, Extending Yamabe flow on complete Riemannian manifolds,, Bull. Sci. Math., 136 (2012), 882. doi: 10.1016/j.bulsci.2012.06.004. Google Scholar [31] R. B. Melrose, Transformation of boundary problems,, Acta Math., 147 (1981), 149. doi: 10.1007/BF02392873. Google Scholar [32] R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem,, Research Notes in Mathematics, (1993). Google Scholar [33] F. Otto and M. Westdickenberg Michael, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227. doi: 10.1137/050622420. Google Scholar [34] J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbbR^n$,, Discrete Contin. Dyn. Syst., 33 (2013), 5407. doi: 10.3934/dcds.2013.33.5407. Google Scholar [35] N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities,, , (). Google Scholar [36] B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics,, Mathematical Topics, (1994). Google Scholar [37] B.-W. Schulze, Boundary Value Problems and Edge Pseudo-Differential Operators,, Microlocal analysis and spectral theory (Lucca, (1996), 165. Google Scholar [38] Y. Shao, A family of parameter-dependent diffeomorphisms acting on function spaces over a Riemannian manifold and applications to geometric flows,, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 45. doi: 10.1007/s00030-014-0275-0. Google Scholar [39] Y. Shao, Singular parabolic equations of second order on manifolds with singularities,, J. Differential Equations, 260 (2016), 1747. doi: 10.1016/j.jde.2015.09.053. Google Scholar [40] Y. Shao, The Yamabe flow on incomplete manifolds,, Submitted. , (). Google Scholar [41] Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds,, J. Evol. Equ., 1 (2014), 211. doi: 10.1007/s00028-014-0218-6. Google Scholar [42] A. E. Shishkov, Waiting time of propagation and the backward motion of interfaces in thin-film flow theory,, Discrete Contin. Dyn. Syst., (2007), 938. Google Scholar [43] H. Triebel, Theory of Function Spaces I,, Birkhäuser Verlag, (1983). doi: 10.1007/978-3-0346-0416-1. Google Scholar [44] J. L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation,, Shape optimization and free boundaries (Montreal, (1990), 347. Google Scholar [45] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, (2007). Google Scholar [46] V. Vespri, Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems,, Ann. Mat. Pura Appl., 155 (1989), 353. doi: 10.1007/BF01765950. Google Scholar [47] X. Xu, Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems,, J. Differential Equations, 252 (2012), 1403. doi: 10.1016/j.jde.2011.08.004. Google Scholar [48] X. Zhu, Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds,, J. Math. Anal. Appl., 402 (2013), 201. doi: 10.1016/j.jmaa.2013.01.018. Google Scholar
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