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Global existence for a partially linear 3D Euler flow. (English) Zbl 1393.35155

Summary: We consider a certain three dimensional Euler flow with infinite energy, which is sometimes called the columnar or two and half dimensional flow. We prove the global smoothness of such flow in \(\mathbb{R}^3\) when the initial data is in some Sobolev or Besov spaces and \(\partial_3 u_3\) is nonnegative.

MSC:

35Q31 Euler equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35M99 Partial differential equations of mixed type and mixed-type systems of partial differential equations
35B65 Smoothness and regularity of solutions to PDEs
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