The generalized Burgers’ equation and the Navier-Stokes equation in $\textbf {R}^ n$ with singular initial data
HTML articles powered by AMS MathViewer
- by Joel D. Avrin PDF
- Proc. Amer. Math. Soc. 101 (1987), 29-40 Request permission
Abstract:
From an abstract theory of Weissler we construct a simple local existence theory for a generalization of Burgers’ equation and the Navier-Stokes equation in the Banach space ${L^p}({{\mathbf {R}}^n})$. Our conditions on $p$ recover the conditions of Giga and Weissler in the latter case except for the borderline situation $p = n$. For the generalized Burgers’ equation our results are apparently new; moreover we show that these local solutions are in fact global solutions in this case. We also obtain results for the generalized Burgers’ equation with ${{\mathbf {R}}^n}$ replaced by a bounded domain $\Omega$ with smooth boundary. Using a somewhat more complex abstract theory of Weissler, we are able to improve on our results found in the case $\Omega = {{\mathbf {R}}^n}$, and also obtain global existence.References
- Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
- J. Avrin, Global existence for generalized transport equations, Mat. Apl. Comput. 4 (1985), no. 1, 67–74 (English, with Portuguese summary). MR 808325
- Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188 Y. Giga, Private communication. —, Solutions for semilinear parabolic equations in ${L^p}$ and regularity of the Navier-Stokes system, J. Differential Equations (to appear).
- Yoshikazu Giga, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 887–910. MR 723454, DOI 10.2977/prims/1195182014 Y. Giga and R. V. Kohn, Asymptotically self-similar blowup of semilinear heat equations (to appear).
- Carl E. Mueller and Fred B. Weissler, Single point blow-up for a general semilinear heat equation, Indiana Univ. Math. J. 34 (1985), no. 4, 881–913. MR 808833, DOI 10.1512/iumj.1985.34.34049 S. Rankin, An abstract semilinear equation which includes Burgers’ equation, talk presented at the Southeastern Atlantic Regional Conference on Differential Equations, Wake Forest University, October 12-13, 1984.
- Fred B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis 32 (1979), no. 3, 277–296. MR 538855, DOI 10.1016/0022-1236(79)90040-5
- Fred B. Weissler, The Navier-Stokes initial value problem in $L^{p}$, Arch. Rational Mech. Anal. 74 (1980), no. 3, 219–230. MR 591222, DOI 10.1007/BF00280539 —, ${L^p}$-energy and blowup for a semilinear heat equation, Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, R. I., 1986, pp. 545-551.
- Fred B. Weissler, Local existence and nonexistence for semilinear parabolic equations in $L^{p}$, Indiana Univ. Math. J. 29 (1980), no. 1, 79–102. MR 554819, DOI 10.1512/iumj.1980.29.29007
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 101 (1987), 29-40
- MSC: Primary 35Q10; Secondary 35K55, 35Q20
- DOI: https://doi.org/10.1090/S0002-9939-1987-0897066-5
- MathSciNet review: 897066