[Paper Review] Continuous Dependence of Cauchy Problem For Nonlinear Schrödinger Equation in $H^{s}$

Wei Dai, Weihua Yang+1 more|arXiv (Cornell University)|Sep 10, 2010

Advanced Mathematical Physics Problems16 references3 citations

TL;DR

This paper establishes continuous dependence of solutions to the nonlinear Schrödinger equation in $H^s( r^N)$ for $s > ext{max}"](1, N/2)$, under $H^s$-subcritical and critical nonlinearities of class $ c(ar{\alpha},s)$. Using Strichartz estimates, fixed-point arguments, and refined Sobolev and Hölder inequalities, it proves local Lipschitz continuity of the solution flow in $H^s$ when the nonlinearity is $C^{[s]+2}$, and continuous dependence otherwise, resolving a long-standing open question on $ar{\varepsilon} = 0$ regularity in the flow map.

ABSTRACT

We consider the Cauchy problem for the nonlinear Schrödinger equation $i \partial_{t}u+ Δu=λ_{0}u+λ_{1}|u|^αu$ in $\mathbb{R}^{N}$, where $λ_{0},λ_{1}\in\mathbb{C}$, in $H^s$ subcritical and critical case: $0

Motivation & Objective

• To resolve the open problem of whether the solution flow to the nonlinear Schrödinger equation is continuous in $H^s$ with $ar{\varepsilon} = 0$, i.e., in the $H^s$-norm itself.
• To establish sharp conditions on the nonlinearity $g_1(u) = \lambda_1 |u|^{\alpha}u$ for continuous dependence in $H^s$ when $1 < s < N/2$ or $s \geq N/2$.
• To extend known results on continuous dependence in $H^{s-\varepsilon}$ to full $H^s$-continuity by refining estimates in Strichartz and Sobolev spaces.
• To provide a complete proof of continuous dependence in $H^s$ for the Cauchy problem under the class $ c(\alpha, s)$, including the critical case $\alpha = 4/(N-2s)$.
• To clarify the role of higher-order regularity ($C^{[s]+2}$) in achieving locally Lipschitz dependence of the solution map on initial data in $H^s$.

Proposed method

• Formulates the Cauchy problem for the nonlinear Schrödinger equation $i\partial_t u + \Delta u = \lambda_0 u + \lambda_1 |u|^\alpha u$ in $\rr^N$, with initial data in $H^s(\rr^N)$, $s > \text{max}\{1, N/2\}$.
• Applies Duhamel’s formula to convert the PDE into an integral equation: $u(t) = e^{it\Delta}\phi - i\int_0^t e^{i(t-\tau)\Delta} g(u(\tau)) d\tau$.
• Uses Strichartz estimates in the space $L^q((-T,T), B^{s}_{r,2}(\rr^N))$ for admissible pairs $(q,r)$ to control the solution norm.
• Employs refined Sobolev and Hölder inequalities to estimate the difference $\|g_1(u_n) - g_1(u)\|_{L^1(I, H^s)}$, relying on the class $\nc(\alpha, s)$ and the behavior of derivatives up to order $[s]+1$.
• Applies Schauder-type estimates and interpolation to control the $L^2$-norm of differences of higher-order derivatives of $g_1(u_n) - g_1(u)$, particularly $\|\Delta_y \partial_x^{[s]}(g_1(u_n) - g_1(u))\|_{L^2}$.
• Uses a fixed-point argument in a ball of $L^\infty((-T,T), H^s)$, with estimates depending on $\|\varphi_n - \varphi\|_{H^s}$, to show $\|u_n - u\|_{L^\infty(I, H^s)} \to 0$ as $n \to \infty$ for small $T$.

Experimental results

Research questions

• RQ1 Can the solution flow of the nonlinear Schrödinger equation be continuous in $H^s(\rr^N)$ for $s > \text{max}\{1, N/2\}$, with the convergence in the $H^s$-norm rather than $H^{s-\varepsilon}$?
• RQ2 What regularity conditions on the nonlinearity $g_1(u) = \lambda_1 |u|^\alpha u$ ensure continuous dependence in $H^s$?
• RQ3 Is the solution map locally Lipschitz continuous in $H^s$ when the nonlinearity is $C^{[s]+2}$, and how does this compare to the $H^{s-\varepsilon}$-continuous case?
• RQ4 How do the estimates of the difference $\|g_1(u_n) - g_1(u)\|_{L^1(I, H^s)}$ behave under the class $\nc(\alpha, s)$, especially when $g_1$ is not $C^{[s]+2}$?
• RQ5 Can the convergence of solutions in $L^\infty((-T,T), H^s)$ be established uniformly over compact time intervals by iterating the local result?

Key findings

• The solution flow is continuous in $H^s(\rr^N)$ for $s > \text{max}\{1, N/2\}$, with convergence in $L^\infty((-T,T), H^s)$, resolving the open problem of $ \varepsilon = 0$ in the standard continuous dependence result.
• If the nonlinearity $g_1$ is in $C^{[s]+2}(\mathbb{C}, \mathbb{C})$, the solution map is locally Lipschitz continuous from $H^s(\rr^N)$ to $L^\infty((-T,T), H^s(\rr^N))$.
• For general $g_1$ of class $\nc(\alpha, s)$, the solution flow is continuous in $H^s$ even without $C^{[s]+2}$ regularity, provided the nonlinearity satisfies the derivative bounds in Definition 1.1.
• The key estimate $\|g_1(u_n) - g_1(u)\|_{L^1(I, H^s)} \leq \varepsilon_n + CT\|u_n - u\|_{L^\infty(I, H^s)}$ holds with $\varepsilon_n \to 0$, which implies convergence after choosing $T$ small enough.
• The proof relies on a careful decomposition of the difference $\Delta_y \partial_x^{[s]}(g_1(u_n) - g_1(u))$ into six terms ($H_1$ to $H_6$), each estimated via Hölder’s inequality, Sobolev embedding, and the class $\nc(\alpha, s)$ conditions.
• The result extends to arbitrary compact subintervals of $(-T_{\min}, T_{\max})$ by iteration, confirming the global continuous dependence of the solution flow in $H^s$.