Year: 2025
Type: preprint
Abstract: In this article, we show the necessary and sufficient conditions for the inequality $\|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}$, where $$\|u\|_{X^{s,b}}:=\|\hat{u}(τ,ξ)\langle ξ\rangle^s\langle τ-ξ^3\rangle^b \|_{L_{τ,ξ}^2}. $$ 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.
Source: arXiv (Cornell University)
Authors: Chen, Jie, Gu, Fan, Guo, Boling
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7077974664
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