Energy method in the partial Fourier space and application to stability problems in the half space

Abstract

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space R n . In this paper, we study half space problems in R n + = R + × R n−1 and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x ∈ R n−1 . For the variable x 1 ∈ R + in the normal direction, we use L 2 space or weighted L 2 space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t → ∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].

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

“…On the other hand, it is useful to apply the energy method in the partial Fourier space to show sharper convergence rate. Indeed the author's paper [14] considered our problem (1.1)-(1.3) with the convex condition (1.5) and obtained the sharper convergence rate of the planar stationary waves. We shall show the result of the paper [14] in detail.…”

“…We use the weighted energy method to the problems (3.2) and (3.3) formulated in the partial Fourier space. Our computation is similar to the one used in [4,10,12,14] and divide into four steps.…”

“…Recently, the author studied half space problems in R n + = R + × R n−1 and developed the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable R n−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]). …”

“…Under the convex condition (1.5), the author in [14] showed the asymptotic stability of the planar stationary wave φ(x 1 ) by proving a sharp decay estimate for the perturbation v(x, t).…”

“…To overcome the difficulty occured by the non-convex condition, we make a good combination of the weighted energy method in partial Fourier space employed in [14] and the anti-derivative method employed in [2,9], and get the desired results. Once we obtain the linear stability results for the problem (1.15), (1.9), (1.10), we may apply this results to the asymptotic stability for the nonlinear problem (1.8)-(1.10).…”