Global Existence for the Two-dimensional Kuramoto-Sivashinsky equation with a Shear Flow

4 Mar 2021  ·  Michele Coti Zelati, Michele Dolce, Yuanyuan Feng, Anna L. Mazzucato ·

We consider the Kuramoto-Sivashinsky equation (KSE) on the two-dimensional torus in the presence of advection by a given background shear flow. Under the assumption that the shear has a finite number of critical points and there are linearly growing modes only in the direction of the shear, we prove global existence of solutions with data in $L^2$, using a bootstrap argument. The initial data can be taken arbitrarily large.

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Analysis of PDEs 35K25, 35K58, 76E06, 76F25