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The generalized Burgers' equation and the Navier-Stokes equation in $ {\bf R}\sp n$ with singular initial data

Author: Joel D. Avrin
Journal: Proc. Amer. Math. Soc. 101 (1987), 29-40
MSC: Primary 35Q10; Secondary 35K55, 35Q20
MathSciNet review: 897066
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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.

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  • [1] R. A. Adams, Sobolev spaces, Academic Press, New York, 1975. MR 0450957 (56:9247)
  • [2] J. Avrin, Global existence for generalized transport equations, Mat. Apl. Comput. 4 (1985), 67-74. MR 808325 (87a:35102)
  • [3] H. Fujita and T. Kato, On the Navier-Stokes initial-value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. MR 0166499 (29:3774)
  • [4] Y. Giga, Private communication.
  • [5] -, Solutions for semilinear parabolic equations in $ {L^p}$ and regularity of the Navier-Stokes system, J. Differential Equations (to appear).
  • [6] -, Weak and strong solutions of the Navier-Stokes initial value problem, Publ. Res. Inst. Math. Sci. 19 (1983), 887-910. MR 723454 (85j:35157)
  • [7] Y. Giga and R. V. Kohn, Asymptotically self-similar blowup of semilinear heat equations (to appear).
  • [8] C. E. Mueller and F. B. Weissler, Single point blow-up for a general semilinear heat equation, IMA preprint series #19, December, 1984. MR 808833 (87a:35023)
  • [9] 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.
  • [10] F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979), 277-296. MR 538855 (80i:47091)
  • [11] -, The Navier-Stokes initial-value problem in $ {L^p}$, Arch. Rational Mech. Anal. 74 (1980), 219-230. MR 591222 (83k:35071)
  • [12] -, $ {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.
  • [13] -, Local existence and nonexistence for semilinear parabolic equations in $ {L^p}$, Indiana Univ. Math. J. 29 (1980), 79-102. MR 554819 (81c:35072)

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