[Paper Review] On Blow-up criterion for the Nonlinear Schrödinger Equation

Dapeng Du, Yifei Wu+1 more|arXiv (Cornell University)|Sep 26, 2013

Advanced Mathematical Physics Problems10 references3 citations

TL;DR

This paper establishes a blow-up criterion for the energy-critical and energy-supercritical nonlinear Schrödinger equation with odd $ p \geq 1 + \frac{4}{N-2} $, proving that solutions with negative initial energy $ E(u_0) < 0 $ must blow up in finite or infinite time. The proof uses a refined virial-type identity and sharp Gagliardo-Nirenberg inequality to show that the $ H^1 $-norm of the solution becomes unbounded, extending previous results beyond the energy-subcritical regime.

ABSTRACT

The blowup is studied for the nonlinear Schrödinger equation $iu_{t}+Δu+ |u|^{p-1}u=0$ with $p$ is odd and $p\ge 1+\frac 4{N-2}$ (the energy-critical or energy-supercritical case). It is shown that the solution with negative energy $E(u_0)&lt;0$ blows up in finite or infinite time. A new proof is also presented for the previous result in \cite{HoRo2}, in which a similar result but more general in a case of energy-subcritical was shown.

Motivation & Objective

• To establish a blow-up criterion for the focusing nonlinear Schrödinger equation in the energy-critical and energy-supercritical regimes where $ p \geq 1 + \frac{4}{N-2} $.
• To extend previous results on negative energy blow-up—previously known only in the energy-subcritical case—to the more singular energy-critical and energy-supercritical regimes.
• To provide a new proof strategy that avoids reliance on finite variance or radial symmetry assumptions, using a refined virial-type functional and sharp Gagliardo-Nirenberg inequality.
• To demonstrate that solutions with negative energy exhibit unbounded $ H^1 $-norm, implying blow-up in finite or infinite time.

Proposed method

• Introduces a modified virial-type functional $ Q(u(t)) $ that captures the energy and gradient norms in a way sensitive to the critical regularity $ s_c = \frac{N}{2} - \frac{2}{p-1} $.
• Uses the sharp Gagliardo-Nirenberg inequality to relate $ L^{p+1} $, $ L^2 $, and $ H^1 $ norms, with the ground state $ Q $ as the sharp constant.
• Applies a contradiction argument assuming $ Q(u(t)) \geq -\delta_0 \|\nabla u(t)\|_{L^2}^2 $ for arbitrarily small $ \delta_0 $, leading to a violation of the initial energy condition.
• Establishes that $ Q(u(t)) < -\delta_0 \|\nabla u(t)\|_{L^2}^2 $ uniformly in time, implying the solution cannot remain bounded in $ H^1 $.
• Uses conservation of mass and energy to derive a uniform lower bound on the gradient norm $ \|\nabla u(t)\|_{L^2} > \epsilon_0 $, ensuring non-compactness.
• Applies a known blow-up theorem (Theorem 2.1) that rules out global $ H^1 $-solutions under such virial-type conditions.

Experimental results

Research questions

• RQ1 Can the blow-up criterion for negative energy initial data in the energy-subcritical NLS be extended to the energy-critical and energy-supercritical regimes?
• RQ2 Does the absence of finite variance or radial symmetry assumptions still allow for blow-up under negative energy?
• RQ3 Can a new virial-type functional be constructed to control the $ H^1 $-norm blow-up in the energy-critical/supercritical case?
• RQ4 Is the sharp Gagliardo-Nirenberg inequality sufficient to derive a uniform lower bound on the gradient norm under negative energy?
• RQ5 What is the role of the critical regularity $ s_c $ in determining the blow-up behavior of the NLS equation?

Key findings

• Solutions to the nonlinear Schrödinger equation with $ p \geq 1 + \frac{4}{N-2} $ and negative initial energy $ E(u_0) < 0 $ must blow up in finite or infinite time.
• The blow-up is characterized by the unboundedness of the $ H^1 $-norm: $ \sup_{t \in (-T_-, T_+)} \|u(t)\|_{H^1} = +\infty $.
• A uniform lower bound $ \|\nabla u(t)\|_{L^2} > \epsilon_0 > 0 $ holds for all $ t $ in the maximal lifespan, preventing global $ H^1 $-regularity.
• The virial-type functional $ Q(u(t)) $ satisfies $ Q(u(t)) < -\delta_0 \|\nabla u(t)\|_{L^2}^2 $ for some $ \delta_0 > 0 $, uniformly in time.
• The proof avoids the need for finite variance or radial symmetry, generalizing earlier results from [9] and [5] to the energy-critical and energy-supercritical regimes.
• The result is established via a contradiction argument using the sharp Gagliardo-Nirenberg inequality and conservation laws, showing that the energy condition $ M(u_0)^{1-s_c} E(u_0)^{s_c} < M(Q)^{1-s_c} E(Q)^{s_c} $ implies blow-up.