Strichartz type estimates of the Airy equation

Abstract

In this article, we show the necessary and sufficient conditions for the inequality $\Vert u{\Vert }_{L_t^q{L}_x^r}\lesssim \Vert u{\Vert }_{X^{s,b}}$, where $$\Vert u{\Vert }_{X^{s,b}}:=\Vert \hat{u}(\tau ,\xi )⟨\xi {⟩}^s⟨\tau -{\xi }^3{⟩}^b{\Vert }_{L_{\tau ,\xi }^2}.$$ Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg--de Vries equations and stochastic Korteweg--de Vries equations.

arXiv comment10 pages