Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Abstract parabolic problems
with critical nonlinearities and applications
to Navier-Stokes and heat equations


Authors: José M. Arrieta and Alexandre N. Carvalho
Journal: Trans. Amer. Math. Soc. 352 (2000), 285-310
MSC (1991): Primary 34G20, 58D25; Secondary 35K05, 35Q30
DOI: https://doi.org/10.1090/S0002-9947-99-02528-3
Published electronically: September 21, 1999
MathSciNet review: 1694278
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a local existence and uniqueness theorem for abstract parabolic problems of the type $\dot x=Ax+f(t,x)$ when the nonlinearity $f$ satisfies certain critical conditions. We apply this abstract result to the Navier-Stokes and heat equations.


References [Enhancements On Off] (What's this?)

  • [AD] Adams, R. Sobolev Spaces, Academic Press, 1975. MR 56:9247
  • [AM1] H. Amann, On abstract parabolic fundamental solutions, J. Math. Soc. Japan 39, (1987), 93-116. MR 88b:34086
  • [AM2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in: Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, vol. 133, Teubner, 1993, pp. 9-126. MR 94m:35153
  • [AM3] H. Amann, Linear and Quasilinear Parabolic Problems. Abstract Linear Theory. Birkhäuser Verlag, 1995. MR 96g:34088
  • [BC] Brezis, H. & Cazenáve, T., A nonlinear heat equation with singular initial data, J. Anal. Math., 68, (1996), 277-304. MR 97f:35092
  • [FK] Fujita, H. & Kato, T., On the Navier-Stokes initial value problem I. Arch. Rat. Mech. Anal. 16 (1964), 269-315. MR 29:3774
  • [Fu] Fujiwara, D., On the asymptotic behavior of the Green operators for elliptic boundary problems and the pure imaginary powers of some second order operators. J. Math. Soc. Japan 21, (1969), 481-521. MR 41:7287
  • [HE] Henry, D., Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics 840, Springer-Verlag, Berlin, (1981). MR 83j:35084
  • [KF] Kato. T. & Fujita, H., On the nonstationary Navier-Stokes system. Rend. Sem. Math. Univ. Padova 32, (1962), 243-260. MR 26:495
  • [LR] Lemarié-Rieusset, P.G. Unicité des solutions des équations de Navier-Stokes, handwritten notes.
  • [Mo] Moser, J., A sharp form of an inequality by N. Trudinger, Indiana U. Math. J., 20, (1971), 1077-1092. MR 46:662
  • [NS] Ni, W.M. & Sacks, P., Singular behavior in nonlinear parabolic equations. Trans. Amer. Math. Soc., 287, (1985), 657-671. MR 86i:35073
  • [PS] Prüss, J and Sohr, H., Imaginary powers of elliptic second order differential operators in $L^p-$spaces. Hiroshima Math. J., 23, (1993), 161-192. MR 94d:47051
  • [S] Seeley, R., Norms and domains of the complex powers $A^z_B$. American Journal of Mathematics, 93 (1971), 299-309. MR 44:4582
  • [Tri] Triebel, H. Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978. MR 80i:46032
  • [Tr] Trudinger, N. S., On imbeddings into Orlicz spaces and some applications, Journal of Mathematics and Mechanics, 17, (1967), 473-483. MR 35:7121
  • [vW] von Wahl, W. The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg & Sohn, Braunschweig, (1985). MR 88a:35195
  • [W1] Weissler, F. B., Semilinear evolution equations in Banach spaces, J. Functional Anal., 32, (1979), 277-296. MR 80i:47091
  • [W2] Weissler, F. B., Local existence and nonexistence for semilinear parabolic equations in $L^p$, Indiana U. Math. J., 29, (1980), 79-102. MR 81e:35072

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 34G20, 58D25, 35K05, 35Q30

Retrieve articles in all journals with MSC (1991): 34G20, 58D25, 35K05, 35Q30


Additional Information

José M. Arrieta
Affiliation: Departamento de Matemática Aplicada, Facultad de Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
Email: arrieta@sunma4.mat.ucm.es

Alexandre N. Carvalho
Affiliation: Departamento de Matemática, Instituto de Ciências Matemáticas de São Carlos, Universidade de São Paulo, C.P. 668, São Carlos, SP. Brazil
Email: andcarva@icmsc.sc.usp.br

DOI: https://doi.org/10.1090/S0002-9947-99-02528-3
Keywords: Abstract parabolic equations, critical nonlinearities, growth conditions, local existence, uniqueness, Navier-Stokes, heat equations.
Received by editor(s): August 6, 1997
Published electronically: September 21, 1999
Additional Notes: The first author’s research was partially supported by FAPESP-SP-Brazil, grant # 1996/3289-4. The second author’s research was partially supported by CNPq-Brazil, grant # 300.889/92-5
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society