Propagation of 𝐿^{𝑞}_{𝑘}-smoothness for solutions of the Euler equation

Ponce, Gustavo

doi:10.1090/S0002-9947-1989-0937250-1

Propagation of $L^ q_ k$-smoothness for solutions of the Euler equation
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Trans. Amer. Math. Soc. 314 (1989), 51-61 Request permission

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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