The generalized Burgers' equation and the Navier-Stokes equation in 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 . Our conditions on recover the conditions of Giga and Weissler in the latter case except for the borderline situation . 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 replaced by a bounded domain with smooth boundary. Using a somewhat more complex abstract theory of Weissler, we are able to improve on our results found in the case , and also obtain global existence.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**J. Avrin,*Global existence for generalized transport equations*, Mat. Apl. Comput.**4**(1985), no. 1, 67–74 (English, with Portuguese summary). MR**808325****[3]**Hiroshi Fujita and Tosio Kato,*On the Navier-Stokes initial value problem. I*, Arch. Rational Mech. Anal.**16**(1964), 269–315. MR**0166499****[4]**Y. Giga, Private communication.**[5]**-,*Solutions for semilinear parabolic equations in**and regularity of the Navier-Stokes system*, J. Differential Equations (to appear).**[6]**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**, 10.2977/prims/1195182014**[7]**Y. Giga and R. V. Kohn,*Asymptotically self-similar blowup of semilinear heat equations*(to appear).**[8]**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**, 10.1512/iumj.1985.34.34049**[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]**Fred B. Weissler,*Semilinear evolution equations in Banach spaces*, J. Funct. Anal.**32**(1979), no. 3, 277–296. MR**538855**, 10.1016/0022-1236(79)90040-5**[11]**Fred B. Weissler,*The Navier-Stokes initial value problem in 𝐿^{𝑝}*, Arch. Rational Mech. Anal.**74**(1980), no. 3, 219–230. MR**591222**, 10.1007/BF00280539**[12]**-,*-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]**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**, 10.1512/iumj.1980.29.29007

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1987-0897066-5

Article copyright:
© Copyright 1987
American Mathematical Society