[Paper Review] Global well-posedness for the radial, defocusing, nonlinear wave equation for $3 < p < 5$

Benjamin Dodson|arXiv (Cornell University)|Oct 5, 2018

Advanced Mathematical Physics Problems11 references7 citations

TL;DR

This paper establishes global well-posedness and scattering for the defocusing nonlinear wave equation in three dimensions with radial initial data in the critical Sobolev space $\dot{H}^{s_c} \times \dot{H}^{s_c-1}$ for $3 < p < 5$. Using the Fourier truncation method and hyperbolic coordinates, the authors prove that solutions exist globally in time and scatter in the critical $L^{2(p-1)}_{t,x}$ norm, with a bound depending only on the initial data size.

ABSTRACT

In this paper we continue the study of the defocusing, energy-subcritical nonlinear wave equation with radial initial data lying in the critical Sobolev space. In this case we prove scattering in the critical norm when $3 < p < 5$.

Motivation & Objective

• To establish global well-posedness and scattering for the defocusing nonlinear wave equation with radial initial data in the critical Sobolev space $\dot{H}^{s_c} \times \dot{H}^{s_c-1}$ for $3 < p < 5$.
• To overcome the lack of a conserved quantity at critical regularity by using the Fourier truncation method to decompose initial data into small and finite-energy parts.
• To prove scattering in the critical $L^{2(p-1)}_{t,x}$ norm by combining hyperbolic coordinates with profile decomposition and Zorn’s lemma.
• To show that the scattering size depends only on the initial data norm, not on its profile, via a perturbative argument and asymptotic orthogonality.
• To extend the method from the $p=4$ case to the full range $3 < p < 5$ using scaling and Strichartz estimates.

Proposed method

• Apply the Fourier truncation method to split initial data into a small $\dot{H}^{s_c} \times \dot{H}^{s_c-1}$-norm part and a finite-energy part.
• Use the hyperbolic coordinate transformation to relate the $L^{2(p-1)}_{t,x}$ norm to the hyperbolic energy, which sandwiches the critical norm.
• Employ the profile decomposition theorem to extract asymptotic profiles from bounded sequences in $\dot{H}^{s_c} \times \dot{H}^{s_c-1}$, with pairwise orthogonal scaling and translation parameters.
• Apply the perturbation lemma to control the $L^{2(p-1)}_{t,x}$ norm of the solution by the norms of the profile solutions and the remainder.
• Use Zorn’s lemma to reduce the proof of uniform boundedness of the $L^{2(p-1)}_{t,x}$ norm to a contradiction argument on a maximal sequence.
• Leverage Strichartz estimates and small data theory to control the asymptotic behavior of profiles, especially in the $\lambda_n^j t_n^j \to \pm\infty$ case, ensuring scattering for each profile.

Experimental results

Research questions

• RQ1 Can global well-posedness and scattering be established for the defocusing nonlinear wave equation in $\mathbb{R}^3$ with radial initial data in the critical Sobolev space when $3 < p < 5$?
• RQ2 How can one prove a uniform $L^{2(p-1)}_{t,x}$ bound for solutions when no conserved quantity exists at the critical regularity?
• RQ3 What role does the hyperbolic coordinate system play in controlling the critical norm and establishing scattering?
• RQ4 Can the profile decomposition and perturbation theory be used to control the $L^{2(p-1)}_{t,x}$ norm of solutions in the absence of a conserved energy at the critical level?
• RQ5 Is it possible to derive a scattering size bound that depends only on the initial data norm, not its profile, in the energy-subcritical regime?

Key findings

• The defocusing nonlinear wave equation is globally well-posed for radial initial data in $\dot{H}^{s_c} \times \dot{H}^{s_c-1}$ with $3 < p < 5$, where $s_c = \frac{3}{2} - \frac{2}{p-1}$.
• Scattering in the critical $L^{2(p-1)}_{t,x}$ norm is established, with the bound satisfying $\|u\|_{L^{2(p-1)}_{t,x}(\mathbb{R} \times \mathbb{R}^3)} \leq f(\|u_0\|_{\dot{H}^{s_c}} + \|u_1\|_{\dot{H}^{s_c-1}})$ for some function $f$.
• The proof relies on decomposing the solution into a finite-energy part and a small-data part via Fourier truncation, ensuring global existence.
• Hyperbolic coordinates are used to relate the $L^{2(p-1)}_{t,x}$ norm to the hyperbolic energy, which bounds the critical norm from above and below.
• The profile decomposition allows control of the $L^{2(p-1)}_{t,x}$ norm by analyzing the asymptotic behavior of profiles under scaling and translation, with orthogonality ensuring no interaction.
• The perturbation lemma and Zorn’s lemma are used to show that the $L^{2(p-1)}_{t,x}$ norm remains uniformly bounded across all bounded initial data sequences, completing the scattering proof.