[Paper Review] Blowup of smooth solutions for general 2-D quasilinear wave equations with small initial data

Bingbing Ding, Ingo Witt+1 more|arXiv (Cornell University)|Jul 8, 2013

Advanced Mathematical Physics Problems16 references3 citations

TL;DR

This paper establishes the finite-time blowup of smooth small-data solutions to general 2D quasilinear wave equations with coefficients depending on both the solution and its gradient. Under the absence of the null condition and a nondegeneracy condition on initial data, it proves the lifespan $ T_
varepsilon $ satisfies $ \lim_{\varepsilon \to 0} \varepsilon \sqrt{T_\varepsilon} = \tau_0 > 0 $, with blowup driven by a singularity in second derivatives at the lifespan time.

ABSTRACT

For the 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}( abla u)\partial_{ij}u=0$ with coefficients independent of the solution $u$, a blowup result for small data solutions has been established in [1,2] provided that the null condition does not hold and a generic nondegeneracy condition on the initial data is fulfilled. In this paper, we are concerned with the more general 2-D quasilinear wave equation $\displaystyle \sum_{i,j=0}^2g_{ij}(u, abla u)\partial_{ij}u=0$ with coefficients that depend simultaneously on $u$ and $ abla u$. When the null condition does not hold and a suitable nondegeneracy condition on the initial data is satisfied, we show that smooth small data solutions blow up in finite time. Furthermore, we derive an explicit expression for the lifespan and establish the blowup mechanism.

Motivation & Objective

• To establish finite-time blowup for smooth small-data solutions of general 2D quasilinear wave equations with coefficient dependence on $ u $ and $ \nabla u $.
• To extend prior blowup results from coefficient-independent to coefficient-dependent cases.
• To derive an explicit asymptotic expression for the lifespan $ T_\varepsilon $ in terms of initial data via Radon transforms and geometric analysis.
• To characterize the blowup mechanism, showing second derivatives blow up like $ 1/(T_\varepsilon - t) $ while first derivatives remain bounded.

Proposed method

• The analysis uses a coordinate transformation to a characteristic variable system $ (s, \theta, \tau) $, where $ s = r - t $, $ \tau = \varepsilon \sqrt{t} $, to localize the blowup near the lifespan time.
• A Nash-Moser-Hörmander iteration scheme is applied to construct $ C^3 $ solutions in a domain $ \mathcal{D}_3 $ near the blowup point, ensuring convergence despite loss of derivatives.
• The lifespan $ T_\varepsilon $ is determined by the infimum of a function $ G_0(\sigma, \theta) $, defined via the Radon transform of initial data and coefficients, leading to $ \tau_0 = \inf G_0(\sigma, \theta) $.
• The nondegeneracy condition (ND) ensures a unique minimum point $ (\sigma_0, \theta_0) $ with positive definite Hessian, guaranteeing the lifespan asymptotics.
• The blowup mechanism is analyzed by estimating the growth of $ \| \nabla_{t,x}^2 u(t, \cdot) \|_{L^\infty} $, showing it behaves like $ C / (T_\varepsilon - t) $ near $ t = T_\varepsilon $.
• The solution is constructed in a neighborhood of the blowup point $ M_\varepsilon = (T_\varepsilon, x_\varepsilon) $, with uniform bounds on $ u $ and $ \nabla u $ in $ C^1 $, confirming continuity up to $ T_\varepsilon $.

Experimental results

Research questions

• RQ1 Under what conditions do smooth small-data solutions to 2D quasilinear wave equations with $ u $- and $ \nabla u $-dependent coefficients blow up in finite time?
• RQ2 What is the precise asymptotic behavior of the lifespan $ T_\varepsilon $ as $ \varepsilon \to 0 $ for such equations?
• RQ3 How does the blowup mechanism differ from ODE-type blowup when coefficients depend on the solution and its gradient?
• RQ4 Can the lifespan be explicitly expressed in terms of initial data and coefficient structure via integral transforms?
• RQ5 What role does the nondegeneracy condition (ND) play in ensuring the lifespan is finite and the blowup is stable under perturbations?

Key findings

• The lifespan $ T_\varepsilon $ satisfies $ \lim_{\varepsilon \to 0} \varepsilon \sqrt{T_\varepsilon} = \tau_0 > 0 $, where $ \tau_0 $ is the infimum of a function $ G_0(\sigma, \theta) $ derived from the Radon transform of initial data.
• The blowup occurs at a unique point $ M_\varepsilon = (T_\varepsilon, x_\varepsilon) $, with second derivatives $ \nabla_{t,x}^2 u $ growing like $ C / (T_\varepsilon - t) $ as $ t \to T_\varepsilon^- $.
• The solution $ u $ remains bounded in $ C^1 $ norm with $ \| u \|_{C^1} \leq C\varepsilon $ and $ \| \nabla_{t,x} u \|_{L^\infty} \leq C\varepsilon $ up to $ T_\varepsilon $, confirming continuity of first derivatives.
• The blowup mechanism is not of ODE type; instead, it arises from the nonlinear coupling in the coefficient structure and the geometry of the initial data.
• The nondegeneracy condition (ND) ensures the existence of a unique minimum point for $ G_0 $, which is essential for the lifespan asymptotics and the blowup profile.
• The function $ G_0(\sigma, \theta) $ is defined via the Radon transform and coefficients, and its infimum determines the leading-order lifespan scaling.