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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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DOI: https://doi.org/10.1090/S0002-9939-1987-0897066-5
Article copyright: © Copyright 1987 American Mathematical Society