Application of the Weighted Energy Method in the Partial Fourier Space to Linearized Viscous Conservation Laws with Non-Convex Condition

Abstract

As you know, the energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. Recently, the author studied half space problems in R+ = R+ × Rn−1 and developed the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable Rn−1. Then the author applied this energy method to the half space problem for linearized viscous conservation laws with convex condition and proved the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞ (see, [14]).