Dissipation enhancement of planar helical flows and applications to three-dimensional Kuramoto-Sivashinsky and Keller-Segel equations

Abstract

We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.

References in corpus (6)

• Suppression of chemotactic explosion by mixing
• Diffusion and mixing in fluid flow via the resolvent estimate
• Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation
• On the well-posedness of an anisotropically-reduced two-dimensional Kuramoto-Sivashinsky equation
• Aggregation equations with fractional diffusion: preventing concentration by mixing
• Global existence for the two-dimensional Kuramoto-Sivashinsky equation with advection