Scale-invariant extinction time estimates for some singular diffusion equations

Yoshikazu Giga; Robert V. Kohn

doi:10.3934/dcds.2011.30.509

Article Contents

2011, Volume 30, Issue 2: 509-535. Doi: 10.3934/dcds.2011.30.509

This issue Previous Article A general theorem on necessary conditions in optimal control Next Article Lipschitz continuity of optimal control and Lagrange multipliers in a problem with mixed and pure state constraints

Scale-invariant extinction time estimates for some singular diffusion equations

Yoshikazu Giga^1, and
Robert V. Kohn^2,

1.
Graduate School of Mathematical Sciences, University of Tokyo, Komaba 3-8-1, Tokyo 153-8914
2.
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012

Received: June 30, 2010

Revised: June 30, 2010

Published: May 2011

Abstract Related Papers Cited by

Abstract

We study three singular parabolic evolutions: the second-order total variation flow, the fourth-order total variation flow, and a fourth-order surface diffusion law. Each has the property that the solution becomes identically zero in finite time. We prove scale-invariant estimates for the extinction time, using a simple argument which combines an energy estimate with a suitable Sobolev-type inequality.

Keywords:

Mathematics Subject Classification: Primary: 35K15, 35K30; Secondary: 35K55, 35B40.

Citation:

Related Papers

Cited by

References

[1]	F. Andreu, V. Caselles, J. I. Diaz and J. M. Mazón, Some qualitative properties for the total variation flow, J. Funct. Anal., 188 (2002), 516-547.doi: 10.1006/jfan.2001.3829.
[2]	F. Andreu-Vaillo, V. Caselles and J. M. Mazón, "Parabolic Quasilinear Equations Minimizing Linear Growth Functionals,'' Progress in Mathematics 223, Birkhauser, Basel, 2004.
[3]	M. Arisawa and Y. Giga, Anisotropic curvature flow in a very thin domain, Indiana Univ. Math. J., 52 (2003), 257-281.doi: 10.1512/iumj.2003.52.2099.
[4]	H. Attouch and A. Damlamian, Application des méthodes de convexité et monotonie à l'étude de certaines équations quasi linéaires, Proc. Roy. Soc. Edinburgh Sect. A, 79 (1977), 107-129.
[5]	V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,'' Noordhoff, Leiden, 1976.
[6]	P. Benilan and M. G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_t-\Delta \varphi (u)=0$, Indiana Univ. Math. J., 30 (1981), 161-177.doi: 10.1512/iumj.1981.30.30014.
[7]	J. Bergh and J. Löfström, "Interpolation Spaces: An Introduction,'' Grundlehren der Mathematischen Wissenschaften 223, Springer-Verlag, Berlin and New York, 1976.
[8]	H. Brezis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert,'' North-Holland Mathematics Studies 5, Notas de Matematica 50. North-Holland, Amsterdam and London; American Elsevier, New York; 1973.
[9]	H. Brezis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis, 9 (1972), 63-74.doi: 10.1016/0022-1236(72)90014-6.
[10]	W.-L. Chan, A. Ramasubramaniam, V. B. Shenoy and E. Chason, Relaxation kinetics of nano-ripples on Cu(001) surfaces, Phys. Rev. B, 70 (2004), 245403.doi: 10.1103/PhysRevB.70.245403.
[11]	E. DiBenedetto, "Degenerate Parabolic Equations,'' Springer-Verlag, New York, 1993.
[12]	I. Ekeland and R. Temam, "Convex Analysis and Variational Problems,'' Studies in Mathematics and its Applications 1, North-Holland, Amsterdam and Oxford; American Elsevier, New York; 1976.
[13]	L. C. Evans and J. Spruck, Motion of level sets by mean curvature III, J. Geom. Anal., 2 (1992), 121-150.
[14]	M.-H. Giga and Y. Giga, Evolving graphs by singular weighted curvature, Arch. Rational Mech. Anal., 141 (1998), 117-198.doi: 10.1007/s002050050075.
[15]	M.-H. Giga and Y. Giga, Very singular diffusion equations - second and fourth order problems, Japan J. Indust. Appl. Math., 27 (2010), 323-345.doi: 10.1007/s13160-010-0020-y.
[16]	M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, in Adv. Stud. Pure Math., 31 (2001), Taniguchi Conference on Mathematics Nara '98, 93-125.
[17]	M.-H. Giga, Y. Giga and J. Saal, "Nonlinear Partial Differential Equations: Asymptotic Behavior of Solutions and Self-Similar Solutions,'' Progress in Nonlinear Differential Equations and Their Applications 79, Birkhauser, Boston, 2010.
[18]	Y. Giga, M. Ohnuma and M.-H. Sato, On the strong maximum principle and the large time behavior of generalized mean curvature flow with the Neumann boundary condition, J. Differential Equations, 154 (1999), 107-131.
[19]	Y. Giga and K. Yama-uchi, On a lower bound for the extinction time of surfaces moved by mean curvature, Calc. Var. Partial Differential Equations, 1 (1993), 417-428.
[20]	E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,'' Monographs in Mathematics 80, Birkhauser Verlag, Basel, 1984.
[21]	J. Hager and H. Spohn, Self-similar morphology and dynamics of periodic surface profiles below the roughening transition, Surf. Sci., 324 (1995), 365-372.doi: 10.1016/0039-6028(94)00771-3.
[22]	Y. Kashima, A subdifferential formulation of fourth order singular diffusion equations, Adv. Math. Sci. Appl., 14 (2004), 49-74.
[23]	R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220.doi: 10.1023/A:1004570921372.
[24]	R. V. Kohn and F. Otto, Upper bounds on coarsening rates, Comm. Math. Phys., 229 (2002), 375-395.doi: 10.1007/s00220-002-0693-4.
[25]	Y. Komura, Nonlinear semi-groups in Hilbert space, J. Math. Soc. Japan, 19 (1967), 493-507.doi: 10.2969/jmsj/01940493.
[26]	A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem, J. Differential Equations, 30 (1978), 340-364.
[27]	D. Margetis and R. V. Kohn, Continuum theory of interacting steps on crystal surfaces in $2+1$ dimensions, Multiscale Model. Simul., 5 (2006), 729-758.doi: 10.1137/06065297X.
[28]	M. V. Ramana Murty, Morphological stability of nanostructures, Phys. Rev. B, 62 (2000), 17004-17011.doi: 10.1103/PhysRevB.62.17004.
[29]	M. Ozdemir and A. Zangwill, Morphological equilibration of a corrugated crystalline surface, Phys. Rev. B, 42 (1990), 5013-5024.doi: 10.1103/PhysRevB.42.5013.
[30]	A. Rettori and J. Villain, Flattening of grooves on a crystal surface: A method of investigation of surface roughness, J. Phys. France, 49 (1988), 257-267.doi: 10.1051/jphys:01988004902025700.
[31]	V. B. Shenoy, A. Ramasubramaniam and L. B. Freund, A variational approach to nonlinear dynamics of nanoscale surface modulations, Surf. Sci., 529 (2003), 365-383.doi: 10.1016/S0039-6028(03)00276-0.
[32]	V. B. Shenoy, A. Ramasubramaniam, H. Ramanarayan, D. T. Tambe, W.-L. Chan and E. Chason, Influence of step-edge barriers on the morphological relaxation of nanoscale ripples on crystal surfaces, Phys. Rev. Lett., 92 (2004), 256101.doi: 10.1103/PhysRevLett.92.256101.
[33]	N. Th. Varopoulos, L. Saloff-Coste and T. Coulhon, "Analysis and Geometry on Groups,'' Cambridge Tracts in Mathematics 100, Cambridge University Press, Cambridge, 1992.
[34]	J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type,'' Oxford Lecture Series in Mathematics and its Applications 33, Oxford University Press, Oxford, 2006.
[35]	J. Watanabe, Approximation of nonlinear problems of a certain type, in "Numerical Analysis of Evolution Equations'' (eds. H. Fujita and M. Yamaguti), Lecture Notes Numer. Appl. Anal. 1, Kinokuniya Book Store, Tokyo, 1979, 147-163.