Local well-posedness for the Davey-Stewartson equation in a generalized Feichtinger algebra
The authors of this interesting paper investigate the initial value problem for the Davey-Stewartson (DS) equations, including elliptic-hyperbolic (EH) and hyperbolic- hyperbolic (HH) cases, \[ \begin{cases} i\partial_tu\pm \partial_{x_1}^{2}u + \partial_{x_2}^{2}u = c_1|u|^2u+c_2u\partial_{x_1}\varphi , \\ \partial_{x_1}^{2}\varphi - \partial_{x_2}^{2}\varphi= \partial_{x_1}|u|^2 , \\ u(0,x_1,x_2) = u_0(x_1,x_2). \end{cases} \tag{*} \] It is assumed that the radiation condition hold: \(\varphi (x_1,x_2)\to 0 \) as \(x_1\pm x_2\to\infty \). Considering the well-posedness of this system in the HH case, it seems that the nonlinearity contains a derivative term which is out of the control of the Strichartz estimates. It turns out that this leads to some difficulty. Define the Feichtinger's modulation space \(M_{p,q}^{s}\) and its anisotropic version \(\breve{M}_{p,q}^{s}\). It is used a frequency-localized smooth effect, maximal functions estimates together with the uniform boundedness of the Schrödinger semigroup \(W (t)=e^{it\triangle_{\pm }}\) in \(M_{1,1}^{s}\) to study the well-posedness of DS equations. The authors focus on local well-posedness in \(\breve{M}_{1,1}^{s}\) with small initial data in \( L^1\). Then the local well-posedness of the DS equations (\(\ast \)) in a generalized Feichtinger algebra is discussed. They show the local existence and uniqueness of the solution in the generalized Feichtinger algebra \(\breve{M}_{1,1}^{s}\) \((s\geq 3/2)\) with sufficiently small initial data in \(\breve{M}_{1,1}^{3/2}(\mathbb{R}^2)\). It is shown that ``the ill-posedness of the solutions in the sense that the solution map is not \(C^3\) if the spatial regularity is below \(\breve{M}_{1,1}^{3/2}\) ''.
scientific article; zbMATH DE number 6493294
scientific article