[Paper Review] Weighted fractional chain rule and nonlinear wave equations with minimal regularity

Kunio Hidano, Jin-Cheng Jiang+2 more|arXiv (Cornell University)|May 22, 2016

Advanced Mathematical Physics Problems36 references3 citations

TL;DR

This paper establishes unconditional local well-posedness for 3D radial quadratic semi-linear wave equations with initial data in the critical regularity range $ s > 3/2 $, filling a long-standing gap between known ill-posedness for $ s < 3/2 $ and well-posedness for $ s \geq 2 $. The key innovation is a new weighted fractional chain rule, enabling sharp Strichartz-type estimates and proving the Glassey conjecture in the radial case with minimal regularity assumptions.

ABSTRACT

We consider the local well-posedness for 3-D quadratic semi-linear wave equations with radial data: $\Box u = a |\partial_t u|^2+b| abla_x u|^2$, $u(0,x)=u_0(x)\in H^{s}_{\mathrm{rad}}$, $\partial_t u(0,x)=u_1(x)\in H^{s-1}_{\mathrm{rad}}$. It has been known that the problem is well-posed for $s\ge 2$ and ill-posed for $s&lt;3/2$. In this paper, we prove unconditional well-posedness up to the scaling invariant regularity, that is to say, for $s&gt;3/2$ and thus fill the gap which was left open for many years. For the purpose, we also obtain a weighted fractional chain rule, which is of independent interest. Our method here also works for a class of nonlinear wave equations with general power type nonlinearities which contain the space-time derivatives of the unknown functions. In particular, we prove the Glassey conjecture in the radial case, with minimal regularity assumption.

Motivation & Objective

• To close the gap in local well-posedness theory for 3D radial quadratic semi-linear wave equations between the known ill-posedness for $ s < 3/2 $ and well-posedness for $ s \geq 2 $.
• To establish unconditional well-posedness in the critical regularity range $ s \in (3/2, 2) $ for radial initial data in $ H^s_{\text{rad}} \times H^{s-1}_{\text{rad}} $.
• To prove the Glassey conjecture in the radial case under minimal regularity assumptions using a new weighted fractional chain rule.
• To derive sharp lifespan estimates for small initial data, showing almost global existence with exponential or polynomial lifespan depending on size.

Proposed method

• Develop a new weighted fractional chain rule for radial functions, which is of independent interest and enables control of nonlinear terms involving derivatives.
• Apply generalized Strichartz estimates with a weight $ w(r) = (1 + r)^{-2\delta} $, where $ \delta = s - 3/2 $, to control the nonlinearities in $ L^2 $-based norms.
• Use a fixed-point argument in a refined function space involving weighted $ L^2 $ norms of $ \partial u $, ensuring continuity and uniqueness of solutions.
• Establish energy and dispersive estimates for the wave operator with radial symmetry, leveraging Hardy’s inequality and weighted $ L^2 $ control.
• Prove unconditional uniqueness by showing that two solutions with the same initial data must agree locally via a small-time contraction argument.
• Extend the result to small data global existence in dimension two by combining generalized Strichartz estimates with $ L^{p-1}_t L^\infty_x $ norms for the nonlinear term.

Experimental results

Research questions

• RQ1 Is the 3D radial quadratic semi-linear wave equation locally well-posed for initial data in $ H^s_{\text{rad}} \times H^{s-1}_{\text{rad}} $ with $ s > 3/2 $, beyond the known $ s \geq 2 $ regime?
• RQ2 Can the Glassey conjecture be proven in the radial case under minimal regularity assumptions?
• RQ3 What is the optimal lifespan of classical solutions in terms of the size of initial data in the critical regularity range?
• RQ4 How can a weighted fractional chain rule be constructed to handle nonlinearities involving space-time derivatives in low regularity settings?

Key findings

• The problem is unconditionally well-posed in $ H^s_{\text{rad}} \times H^{s-1}_{\text{rad}} $ for all $ s > 3/2 $, with a unique solution in $ L^\infty_T H^s \cap \text{Lip}_T H^{s-1} $.
• The lifespan $ T_\varepsilon $ of solutions satisfies $ T_\varepsilon = \exp(c\varepsilon^{-1}) $ for $ \varepsilon < 1 $, and $ T_\varepsilon = c\varepsilon^{-1/(s-3/2)} $ for $ \varepsilon \geq 1 $, with $ \varepsilon $ measuring initial data size.
• The solution satisfies improved regularity: $ u \in C([0,T]; H^s) \cap C^1([0,T]; H^{s-1}) $, and $ r^{-1/2 + \delta} \langle r \rangle^{-\delta} \partial u \in L^2([0,T] \times \mathbb{R}^3) $ with $ \delta = s - 3/2 $.
• The weighted fractional chain rule is established and used to control $ \| \partial (|u|^p) \| $ in $ L^2 $-based norms, enabling the proof of Strichartz estimates in low regularity.
• The Glassey conjecture is confirmed in the radial case for $ s > 3/2 $, with minimal regularity, by proving well-posedness and lifespan estimates.
• Small data global existence is established in dimension two for $ p > 5 $, with critical regularity $ s_c = 2 - 1/(p-1) \in (1,2) $, using generalized Strichartz estimates and contraction in $ C_t H^{s_c} \cap C_t^1 H^{s_c - 1} $.