Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type

YAO Feng-ping; ZHOU Shu-lin

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2007 > 28(11): 1340-1352

YAO Feng-ping, ZHOU Shu-lin. Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1340-1352.

Citation:

YAO Feng-ping, ZHOU Shu-lin. Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1340-1352.

Citation:

YAO Feng-ping, ZHOU Shu-lin. Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type[J]. Applied Mathematics and Mechanics, 2007, 28(11): 1340-1352.

PDF( 501 KB)

Schauder Estimates for the Parabolic Equation of the Bi-Harmonic Type

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

Received Date: 2006-10-23
Rev Recd Date: 2007-08-20
Publish Date: 2007-11-15

Abstract

Abstract

Global Schauder estimates for the initial-value parabolic problem of the bi-harmonic type were proved. The existence and uniqueness of the solutions in the suitable space were obtained. Similarly to the second-order case, a fomal expression of solutions by the Fourier transform was obtained. Then the regularity, uniqueness, existence of solutions using the potential theory and approximation argument were got. The approach is simple and straightforward.
- bi-harmonic,
- parabolic,
- Schauder estimate,
- existence,
- uniqueness

FullText(HTML)

References(25)

References

[1]	Schauder J.ber lineare elliptische Differentialgleichungen zweiter Ordnung[J].Math Z,1934,38(1):257-282. doi: 10.1007/BF01170635
[2]	Schauder J.Numerische Abschdtzungen in elliptischen linearen Differentialgleichungen[J].Studia Math,1934,5(1):34-42.
[3]	Campanato S.Propriet di una famiglia di spazi funzionali[J].Ann Scuola Norm Sup Pisa,1964,18(3):137-160.
[4]	Trudinger N S.A new approach to the Schauder estimates for linear elliptic equations[J].Proc Centre Math Anal Austral Nat Univ,1986,14:52-59.
[5]	Caffarelli L A.Interior a priori estimates for solutions of fully nonlinear equations[J].Ann Math,1989,130(1):189-213. doi: 10.2307/1971480
[6]	Ciliberto C.Formule di maggiorazione e teoremi di esistenza per le soluzioni delle equazioni paraboliche in due variabili[J].Ricerche Mat,1954,3(1):40-75.
[7]	Campanato S.Equazioni paraboliche del secondo ordine e spazi L[KG*2]. 2,λ(Ω;δ)[J].Ann Math Pura Appl,1966,73(4):55-102.
[8]	Simon L.Schauder estimates by scaling[J].Calc Var PDE,1997,5(5):391-407. doi: 10.1007/s005260050072
[9]	WANG Li-he.On the regularity theory of fully nonlinear parabolic equations Ⅱ[J].Comm Pure Appl Math,1992,45(2):141-178. doi: 10.1002/cpa.3160450202
[10]	Friedman A.Partial Differential Equations of Parabolic Type[M].Englewood Cliffs,NJ:Prentice-Hall Inc,1964.
[11]	Ladyzenskaja O A,Solonnikov V A,Uralceva N N.Linear and Quasilinear Equations of Parabolic Type[M].Providence, RI:American Mathematical Society,1968.
[12]	Lorenzi L.Schauder estimates for degenerate elliptic and parabolic problems with unbounded coefficients in RN[J].Differential Internat Equations,2005,18(5):531-566.
[13]	Lunardi A.Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in RN[J].Studia Math,1998,128(2):171-198.
[14]	Lunardi A.Schauder estimates for a class of degenerateelliptic and parabolic operators with unbounded coefficients in RN[J].Annali della Scuola Normale Superiore Pisa,1997,24(4):133-164.
[15]	腾振寰.抛物型方程一般边界问题解的先验估计[J].数学进展,1965,4(3):334-386.
[16]	王柔怀.关于一般抛物型边值问题的Schauder估计[J].吉林大学自然科学学报,1964,2(1):35-64.
[17]	Solonnikov V A.On boundary value problems for linear parabolic systems of differential equations of general form[J].Trudy Mat Inst Steklov,1965,83:3-163.
[18]	Cahn J W,Hilliard J E.Free energy of nonuniform system I. Interfacial free energy[J].J Chem Phys,1958,28(2):258-367. doi: 10.1063/1.1744102
[19]	Alikakos N D,Fusco G.Slow dynamics for the Cahn-Hilliard equation in higher space dimensions: the motion of bubbles[J].Arch Rat Mech Anal,1998,141(1):1-61. doi: 10.1007/s002050050072
[20]	Rossi R.On two classes of generalized viscous Cahn-Hilliard equations[J].Comm Pure Appl Anal,2005,4(2):405-430. doi: 10.3934/cpaa.2005.4.405
[21]	Kwembe T A.Existence and uniqueness of global solutions for the parabolic equation of the bi-harmonic type[J].Nonlinear Anal,2001,47(2):1321-1332. doi: 10.1016/S0362-546X(01)00268-1
[22]	XU Meng,ZHOU Shu-lin.Existence and uniqueness of weak solutions for a generalized thin film equation[J].Nonlinear Anal,2005,60(4):755-774. doi: 10.1016/j.na.2004.01.013
[23]	YIN Jing-xue,LIU Chang-chun.Regularity of solutions of the Cahn-Hilliard equation with concentration dependent mobility[J].Nonlinear Anal,2001,45(5):543-554. doi: 10.1016/S0362-546X(99)00406-X
[24]	DiBenedetto E.Partial Differential Equations[M].Boston-Basel-Berlin:Birkhuser,1995.
[25]	CHEN Ya-zhe,WU Lan-cheng.Second Order Elliptic Partial Differential Equations and Elliptic Systems[M].Providence,RI:American Mathematical Society,1998.