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Propagation of $ L\sp q\sb k$-smoothness for solutions of the Euler equation

Author: Gustavo Ponce
Journal: Trans. Amer. Math. Soc. 314 (1989), 51-61
MSC: Primary 35B65; Secondary 35Q10, 76C10
MathSciNet review: 937250
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Abstract: The motion of an ideal incompressible fluid is described by a system of partial differential equations known as the Euler equation. Considering the initial value problem for this equation, we prove that in a classical solution the $ L_k^q$-regularity of the data propagates along the fluid lines. Our method consists of combining properties of the $ \varepsilon $-approximate solution with $ {L^q}$-energy estimates and simple results of classical singular integral operators. In particular, for the two-dimensional case we present an elementary proof.

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Keywords: Euler equations, $ \varepsilon $-approximate solutions, $ {L^q}$-energy estimates
Article copyright: © Copyright 1989 American Mathematical Society