The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity

TL;DR

This work analyzes the Cauchy problem for the 3D energy-critical INLS equation $i\partial_t u + \Delta u = \pm |x|^{-\alpha} |u|^{4-2\alpha} u$ with strong singularity $3/2 \le \alpha < 2$. By developing extended inhomogeneous Strichartz estimates in weighted spaces and performing time-localized, dyadic bilinear analysis, the authors expand the admissible exponent range and control the singular nonlinear term to prove local well-posedness and small-data global well-posedness for $3/2 \le \alpha < 11/6$. They construct tailored function spaces with weighted norms and demonstrate contraction mappings within these spaces, leveraging the smoothing effect of the Schrödinger flow. The paper also establishes sharp necessary conditions for the weighted Strichartz estimates via stationary-phase arguments, delineating the limits of the weighted approach. Overall, the results advance the well-posedness theory for the energy-critical INLS with strong singularity by exploiting weighted spaces and refined Strichartz machinery.

Abstract

In this paper, we study the Cauchy problem for the 3D energy-critical inhomogeneous nonlinear Schrödinger equation(INLS) $$i\partial_{t}u+Δu=\pm|x|^{-α}|u|^{4-2α}u$$ with strong singularity $3/2\leq α<2$. The well-posedness problem is well-understood for $0<α<3/2$, but the case $3/2\leq α<2$ has remained open so far. We address the local/small data global well-posedness result for $3/2\leq α<11/6$ by improving the inhomogeneous Strichartz estimates on the weighted space.

Figures (2)

• Figure 1: The validity of $(1/r, 1/\tilde{r})$ on \ref{['inhomo']}.
• Figure 2: The validity of $(1/b, 1/\tilde{b})$ on \ref{['eq2']}.

Theorems & Definitions (17)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Remark 1.6
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof