[Paper Review] Global well-posedness for the defocusing mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$ at $L^2$ regularity

Chenjie Fan, Weijun Xu|arXiv (Cornell University)|Oct 18, 2018

Advanced Mathematical Physics Problems3 references9 citations

TL;DR

This paper establishes global well-posedness for the defocusing mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$ at $L^2$ regularity using a truncated subcritical approximation and limit passage. It proves existence, uniqueness, and pathwise mass conservation for solutions with arbitrary $L^2$ initial data, under a sufficiently regular noise assumption, and establishes stability under initial data perturbations.

ABSTRACT

We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schrödinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem. With the presence of truncation, we construct the solution to the critical equation as the limit of subcritical solutions. We then obtain uniform bounds on the solutions to the truncated critical problems that allow us to remove truncation in the limit.

Motivation & Objective

• To establish global existence and stability of solutions to the defocusing mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$ with arbitrary $L^2$ initial data.
• To overcome the breakdown of standard fixed-point arguments due to the interplay between nonlinearity and multiplicative noise in the natural solution space.
• To construct a solution via a limiting procedure from truncated subcritical problems, ensuring uniform bounds that allow removal of the truncation.
• To prove pathwise mass conservation and stability of the solution under perturbations of the initial data.
• To extend the deterministic mass-critical theory to the stochastic setting with Itło-Stratonovich correction and regular noise.

Proposed method

• Construct solutions to a truncated subcritical version of the SNLS equation using a smooth cutoff function $\theta_m$ based on the $\mathcal{X}_2$-norm of the solution.
• Apply a priori estimates, including dispersive and Strichartz inequalities, to derive uniform bounds on the truncated solutions in $L^\rho(\Omega; \mathcal{X}(0,T))$ for $\rho > 5$.
• Use the Burkholder-Davis-Gundy inequality and stochastic integration theory to control the Itło stochastic integral term in the Duhamel formulation.
• Prove convergence of the truncated solutions $u_m$ to a limit $u$ in $L^\rho(\Omega; \mathcal{X}(0,T))$ by showing the sequence is Cauchy in this space.
• Establish the limit $u$ satisfies the Duhamel formula in Itło form, including the Itło-Stratonovich correction term $F_\Phi u$, via dominated convergence and Strichartz estimates.
• Use stability estimates for the subcritical problem and convergence in $L^\rho$-norm to prove continuous dependence on initial data.

Experimental results

Research questions

• RQ1 Can global well-posedness be established for the mass-critical SNLS on $\mathbb{R}$ at $L^2$ regularity despite the failure of standard fixed-point methods due to noise and nonlinearity?
• RQ2 How can one construct a solution to the critical SNLS equation when the noise and nonlinearity prevent direct application of deterministic methods?
• RQ3 Does the solution to the SNLS preserve the $L^2$-norm pathwise, and is this preserved under the Itło-Stratonovich correction?
• RQ4 Can uniform bounds on truncated subcritical solutions be derived to allow removal of the truncation in the limit?
• RQ5 Is the solution stable under small perturbations of the initial data in the $L^2$-sense, and can this stability be quantified?

Key findings

• Global existence and uniqueness of a solution $u$ to the mass-critical SNLS on $\mathbb{R}$ is established for arbitrary $L^2$ initial data in the $L^\infty(\Omega; L^2(\mathbb{R}))$ class.
• The solution satisfies the Duhamel formula in Itło form with the correction term $F_\Phi u$, and lies in $L^{\rho_0}(\Omega; \mathcal{X}(0,T))$ for every $T>0$ and $\rho_0 > 5$, with a uniform bound depending only on $T$, $\rho_0$, and $\|u_0\|_{L^\infty(\Omega; L^2)}$.
• Pathwise mass conservation holds: $\|u(t)\|_{L^2} = \|u_0\|_{L^2}$ for all $t \in [0,T]$ almost surely.
• The solution is stable under initial data perturbations: for every $M>0$ and $\delta>0$, there exists $\kappa>0$ such that $\|u_0 - v_0\|_{L^\infty(\Omega; L^2)} < \kappa$ implies $\|u - v\|_{L^{\rho_0}(\Omega; \mathcal{X}(0,T))} < \delta$, with $\kappa$ depending only on $M$, $\delta$, $T$, and $\rho_0$.
• The limit of the truncated solutions $u_m$ exists in $L^{\rho_0}(\Omega; \mathcal{X}(0,T))$ and satisfies the full SNLS equation in the sense of the Duhamel formula.
• The construction relies on uniform bounds and convergence in $L^\rho$-norms, with the key estimate $\|u_m - u_{m'}\|_{L^{\rho_0}(\Omega; \mathcal{X}(0,1))} < \delta$ for large $m, m'$, ensuring Cauchy convergence.