New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations

Michael Goldman (1) , Marc Josien (2) , Felix Otto (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 MPI-MiS - Max-Planck-Institut für Mathematik in den Naturwissenschaften

We give a substantially simplified proof of near-optimal estimate on the Kuramoto-Sivashinsky equation from [F. Otto, "Optimal bounds on the Kuramoto-Sivashinsky equation", JFA 2009], at the same time slightly improving the result. The result in the above cited paper relied on two ingredients: a regularity estimate for capillary Burgers and an a novel priori estimate for the inhomogeneous inviscid Burgers equation, which works out that in many ways the {\it conservative transport} nonlinearity acts as a t coercive term. It is the proof of the second ingredient that we substantially simplify by proving a modified K\'arm\'an-Howarth-Monin identity for solutions of the inhomogeneous inviscid Burgers equation. This gives a new interpretation of the results obtained in [F. Golse, B. Perthame "Optimal regularizing effect for scalar conservation laws", Rev. Mat. Iber., 2013].

Mots clés: Kuramoto-Sivashinsky equation; Burgers equation

Domaines: Equations aux dérivées partielles [math.AP]

Michael Goldman, Marc Josien, Felix Otto. New bounds for the inhomogenous Burgers and the Kuramoto-Sivashinsky equations. Communications in Partial Differential Equations, 2015. ⟨hal-01133693⟩