The interior-boundary Strichartz estimate for the Schrödinger equation on the half line revisited

13 Jan 2021  ·  Türker Özsarı ·

We introduce an alternate yet a simple Fokas method (aka UTM) based approach for proving the interior-boundary Strichartz estimate for solutions of the initial boundary value problem (ibvp) for the Schr\"odinger equation posed on the right half line. We utilize the solution representation formula obtained through the Fokas method and a few basic tools from the oscillatory integral theory... Representation formula for solutions of the ibvp splits into two parts, one of which is analyzed via the theory of oscillatory integrals, whereas the second part nicely transforms, via a time-to-space switch, into a Cauchy problem for which Strichartz estimates are already known. The obtained estimate implies the local wellposedness of low regularity solutions for the associated nonlinear ibvp, which was previously treated by other powerful however technical methods. We in particular extend the recent Fokas method based $L_t^\infty H_x^s(\mathbb{R}_+)$ estimates to Fokas method based $L_t^\lambda W_x^{s,r}(\mathbb{R}_+)$ estimates for any admissible pair $(\lambda,r)$. This implies that UTM formulas are capable of defining weak solutions of NLS also below the Banach algebra threshold $C_tH_x^\frac{1}{2}(\mathbb{R}_+)$. The approach is uniform and can be easily extended to a large class of ibvps and boundary values associated with dispersive equations, this is all thanks to the nice oscillatory kernels of the associated representation formulas. In particular, Strichartz estimates for the Neumann problem are new here. read more

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