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 and 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, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

H. Amann, Elliptic operators with infinite-dimensional state spaces, J. Evol. Equ., 1 (2001), 143-188. doi: 10.1007/PL00001367.

[3]

H. Amann, Function spaces on singular manifolds, Math. Nachr., 286 (2013), 436-475. doi: 10.1002/mana.201100157.

[4]

H. Amann, Anisotropic function spaces on singular manifolds,, , ().  doi: 10.1002/mana.201100157.

[5]

H. Amann, Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems, Recent Developments of Mathematical Fluid Mechanics, Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (Eds.) Series: Advances in Mathematical Fluid Mechanics, Birkhaeuser-Verlag, 2016. doi: 10.1007/978-3-0348-0939-9_4.

[6]

H. Amann, Uniformly Regular and Singular Riemannian Manifolds, Elliptic and parabolic equations, Springer Proc. Math. Stat., 119 (2015), Springer, Cham 1-43. doi: 10.1007/978-3-319-12547-3_1.

[7]

H. Amann, Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and Hölder Spaces on Uniformly Regular Riemannian Manifolds,, , (). 

[8]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.

[9]

E. Bahuaud and B. Vertman, Yamabe flow on manifolds with edges, Math. Nachr., 287 (2014), 127-159. doi: 10.1002/mana.201200210.

[10]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.

[11]

M. Boutat, S. Hilout, J.-E. Rakotoson and J.-M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal., 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028.

[12]

F. E. Browder, Analyticity and partial differential equations I, Amer. J. Math., 84 (1962), 666-710. doi: 10.2307/2372872.

[13]

D. Chang, N. Habal and B.-W. Schulze, Quantisation on a manifold with singular edge, J. Pseudo-Differ. Oper. Appl., 4 (2013), 317-343. doi: 10.1007/s11868-013-0077-x.

[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, 1250009, 12 pp (2012). doi: 10.1142/S0129167X11007525.

[15]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67. doi: 10.1007/PL00001364.

[16]

R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437-463.

[17]

G. Da Prato and P. Grisvard, Equations d'évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl., (4) 120 (1979), 329-396. doi: 10.1007/BF02411952.

[18]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965. doi: 10.1090/S0894-0347-98-00277-X.

[19]

S. A. J. Dekkers, A comparison theorem for solutions of degenerate parabolic equations on manifolds, Proc. Roy. Soc. Edinburgh Sect., A 138 (2008), 755-767. doi: 10.1017/S0308210505000880.

[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), 385-415. doi: 10.1007/BF03167255.

[21]

E. Di Benedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[22]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-1584-8.

[23]

M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., 289 (2016), 232-242. doi: 10.1002/mana.201400354.

[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), 103-116.

[25]

L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces Free Bound., 8 (2006), 111-129. doi: 10.4171/IFB/137.

[26]

G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. Inst. H. Poincaré Anal., Non Linéaire 21 (2004), 255-269. doi: 10.1016/j.anihpc.2003.02.002.

[27]

J. R. King, Two generalisations of the thin film equation, Math. Comput. Modelling, 34 (2001), 737-756. doi: 10.1016/S0895-7177(01)00095-4.

[28]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152. doi: 10.1512/iumj.2008.57.3349.

[29]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[30]

L. Ma, L. Cheng and A. Zhu, Extending Yamabe flow on complete Riemannian manifolds, Bull. Sci. Math., 136 (2012), 882-891. doi: 10.1016/j.bulsci.2012.06.004.

[31]

R. B. Melrose, Transformation of boundary problems, Acta Math., 147 (1981), 149-236. doi: 10.1007/BF02392873.

[32]

R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993.

[33]

F. Otto and M. Westdickenberg Michael, Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., 37 (2005), 1227-1255. doi: 10.1137/050622420.

[34]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbbR^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428. doi: 10.3934/dcds.2013.33.5407.

[35]

N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities,, , (). 

[36]

B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics, Mathematical Topics, 4. Akademie Verlag, Berlin, 1994.

[37]

B.-W. Schulze, Boundary Value Problems and Edge Pseudo-Differential Operators, Microlocal analysis and spectral theory (Lucca, 1996), 165-226, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, Kluwer Acad. Publ., Dordrecht, 1997.

[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), 45-85. doi: 10.1007/s00030-014-0275-0.

[39]

Y. Shao, Singular parabolic equations of second order on manifolds with singularities, J. Differential Equations, 260 (2016), 1747-1800. doi: 10.1016/j.jde.2015.09.053.

[40]

Y. Shao, The Yamabe flow on incomplete manifolds,, Submitted. , (). 

[41]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 1 (2014), 211-248. doi: 10.1007/s00028-014-0218-6.

[42]

A. E. Shishkov, Waiting time of propagation and the backward motion of interfaces in thin-film flow theory, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 938-945.

[43]

H. Triebel, Theory of Function Spaces I, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[44]

J. L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation, Shape optimization and free boundaries (Montreal, PQ, 1990), 347-389, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, Kluwer Acad. Publ., Dordrecht, 1992.

[45]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[46]

V. Vespri, Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems, Ann. Mat. Pura Appl., (4) 155 (1989), 353-388. doi: 10.1007/BF01765950.

[47]

X. Xu, Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems, J. Differential Equations, 252 (2012), 1403-1420. doi: 10.1016/j.jde.2011.08.004.

[48]

X. Zhu, Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds, J. Math. Anal. Appl., 402 (2013), 201-206. doi: 10.1016/j.jmaa.2013.01.018.

show all references

References:
[1]

H. Amann, Linear and Quasilinear Parabolic Problems: Volume I., Abstract linear theory. Monographs in Mathematics, 89. Birkhäuser Boston, Inc., Boston, MA, 1995. doi: 10.1007/978-3-0348-9221-6.

[2]

H. Amann, Elliptic operators with infinite-dimensional state spaces, J. Evol. Equ., 1 (2001), 143-188. doi: 10.1007/PL00001367.

[3]

H. Amann, Function spaces on singular manifolds, Math. Nachr., 286 (2013), 436-475. doi: 10.1002/mana.201100157.

[4]

H. Amann, Anisotropic function spaces on singular manifolds,, , ().  doi: 10.1002/mana.201100157.

[5]

H. Amann, Parabolic equations on uniformly regular Riemannian manifolds and degenerate initial boundary value problems, Recent Developments of Mathematical Fluid Mechanics, Amann, H., Giga, Y., Kozono, H., Okamoto, H., Yamazaki, M. (Eds.) Series: Advances in Mathematical Fluid Mechanics, Birkhaeuser-Verlag, 2016. doi: 10.1007/978-3-0348-0939-9_4.

[6]

H. Amann, Uniformly Regular and Singular Riemannian Manifolds, Elliptic and parabolic equations, Springer Proc. Math. Stat., 119 (2015), Springer, Cham 1-43. doi: 10.1007/978-3-319-12547-3_1.

[7]

H. Amann, Cauchy Problems for Parabolic Equations in Sobolev-Slobodeckii and Hölder Spaces on Uniformly Regular Riemannian Manifolds,, , (). 

[8]

S. B. Angenent, Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), 91-107. doi: 10.1017/S0308210500024598.

[9]

E. Bahuaud and B. Vertman, Yamabe flow on manifolds with edges, Math. Nachr., 287 (2014), 127-159. doi: 10.1002/mana.201200210.

[10]

F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations, 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y.

[11]

M. Boutat, S. Hilout, J.-E. Rakotoson and J.-M. Rakotoson, A generalized thin-film equation in multidimensional space, Nonlinear Anal., 69 (2008), 1268-1286. doi: 10.1016/j.na.2007.06.028.

[12]

F. E. Browder, Analyticity and partial differential equations I, Amer. J. Math., 84 (1962), 666-710. doi: 10.2307/2372872.

[13]

D. Chang, N. Habal and B.-W. Schulze, Quantisation on a manifold with singular edge, J. Pseudo-Differ. Oper. Appl., 4 (2013), 317-343. doi: 10.1007/s11868-013-0077-x.

[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, 1250009, 12 pp (2012). doi: 10.1142/S0129167X11007525.

[15]

P. Clément and G. Simonett, Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations, J. Evol. Equ., 1 (2001), 39-67. doi: 10.1007/PL00001364.

[16]

R. Dal Passo, L. Giacomelli and G. Grün, A waiting time phenomenon for thin film equations, Ann. Scuola Norm. Sup. Pisa, 30 (2001), 437-463.

[17]

G. Da Prato and P. Grisvard, Equations d'évolution abstraites non linéaires de type parabolique, Ann. Mat. Pura Appl., (4) 120 (1979), 329-396. doi: 10.1007/BF02411952.

[18]

P. Daskalopoulos and R. Hamilton, Regularity of the free boundary for the porous medium equation, J. Amer. Math. Soc., 11 (1998), 899-965. doi: 10.1090/S0894-0347-98-00277-X.

[19]

S. A. J. Dekkers, A comparison theorem for solutions of degenerate parabolic equations on manifolds, Proc. Roy. Soc. Edinburgh Sect., A 138 (2008), 755-767. doi: 10.1017/S0308210505000880.

[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), 385-415. doi: 10.1007/BF03167255.

[21]

E. Di Benedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.

[22]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics. Springer, New York, 2012. doi: 10.1007/978-1-4614-1584-8.

[23]

M. Disconzi, Y. Shao and G. Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr., 289 (2016), 232-242. doi: 10.1002/mana.201400354.

[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), 103-116.

[25]

L. Giacomelli and G. Grün, Lower bounds on waiting times for degenerate parabolic equations and systems, Interfaces Free Bound., 8 (2006), 111-129. doi: 10.4171/IFB/137.

[26]

G. Grün, Droplet spreading under weak slippage: the waiting time phenomenon, Ann. Inst. H. Poincaré Anal., Non Linéaire 21 (2004), 255-269. doi: 10.1016/j.anihpc.2003.02.002.

[27]

J. R. King, Two generalisations of the thin film equation, Math. Comput. Modelling, 34 (2001), 737-756. doi: 10.1016/S0895-7177(01)00095-4.

[28]

A. Kiselev, R. Shterenberg and A. Zlatoš, Relaxation enhancement by time-periodic flows, Indiana Univ. Math. J., 57 (2008), 2137-2152. doi: 10.1512/iumj.2008.57.3349.

[29]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel, 1995. doi: 10.1007/978-3-0348-9234-6.

[30]

L. Ma, L. Cheng and A. Zhu, Extending Yamabe flow on complete Riemannian manifolds, Bull. Sci. Math., 136 (2012), 882-891. doi: 10.1016/j.bulsci.2012.06.004.

[31]

R. B. Melrose, Transformation of boundary problems, Acta Math., 147 (1981), 149-236. doi: 10.1007/BF02392873.

[32]

R. B. Melrose, The Atiyah-Patodi-Singer Index Theorem, Research Notes in Mathematics, 4. A K Peters, Ltd., Wellesley, MA, 1993.

[33]

F. Otto and M. Westdickenberg Michael, Eulerian calculus for the contraction in the Wasserstein distance, SIAM J. Math. Anal., 37 (2005), 1227-1255. doi: 10.1137/050622420.

[34]

J. Prüss and G. Simonett, On the manifold of closed hypersurfaces in $\mathbbR^n$, Discrete Contin. Dyn. Syst., 33 (2013), 5407-5428. doi: 10.3934/dcds.2013.33.5407.

[35]

N. Roidos and E. Schrohe, Existence and maximal $L^p$-regularity of solutions for the porous medium equation on manifolds with conical singularities,, , (). 

[36]

B.-W. Schulze, Pseudo-differential Boundary Value Problems, Conical Singularities, and Asymptotics, Mathematical Topics, 4. Akademie Verlag, Berlin, 1994.

[37]

B.-W. Schulze, Boundary Value Problems and Edge Pseudo-Differential Operators, Microlocal analysis and spectral theory (Lucca, 1996), 165-226, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 490, Kluwer Acad. Publ., Dordrecht, 1997.

[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), 45-85. doi: 10.1007/s00030-014-0275-0.

[39]

Y. Shao, Singular parabolic equations of second order on manifolds with singularities, J. Differential Equations, 260 (2016), 1747-1800. doi: 10.1016/j.jde.2015.09.053.

[40]

Y. Shao, The Yamabe flow on incomplete manifolds,, Submitted. , (). 

[41]

Y. Shao and G. Simonett, Continuous maximal regularity on uniformly regular Riemannian manifolds, J. Evol. Equ., 1 (2014), 211-248. doi: 10.1007/s00028-014-0218-6.

[42]

A. E. Shishkov, Waiting time of propagation and the backward motion of interfaces in thin-film flow theory, Discrete Contin. Dyn. Syst., 2007, Dynamical Systems and Differential Equations. Proceedings of the 6th AIMS International Conference, suppl., 938-945.

[43]

H. Triebel, Theory of Function Spaces I, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[44]

J. L. Vázquez, An Introduction to the Mathematical Theory of the Porous Medium Equation, Shape optimization and free boundaries (Montreal, PQ, 1990), 347-389, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 380, Kluwer Acad. Publ., Dordrecht, 1992.

[45]

J. L. Vázquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[46]

V. Vespri, Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems, Ann. Mat. Pura Appl., (4) 155 (1989), 353-388. doi: 10.1007/BF01765950.

[47]

X. Xu, Gradient estimates for $u_t=\Delta F(u)$ on manifolds and some Liouville-type theorems, J. Differential Equations, 252 (2012), 1403-1420. doi: 10.1016/j.jde.2011.08.004.

[48]

X. Zhu, Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds, J. Math. Anal. Appl., 402 (2013), 201-206. doi: 10.1016/j.jmaa.2013.01.018.

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