[Paper Review] Stable blow up dynamics for the 1-corotational energy critical harmonic heat flow

Pierre Raphaël, Rémi Schweyer|arXiv (Cornell University)|Jun 5, 2011

Advanced Mathematical Physics ProblemsMathematics27 references103 citations

TL;DR

This paper establishes the existence of a stable, finite-time blow-up regime for the 1-corotational energy-critical harmonic heat flow from $ ^2$ into a smooth revolution surface in $ ^3$, proving that solutions with smooth, localized initial data arbitrarily close to the ground state harmonic map develop a singularity via concentration of energy in a universal bubble. The blow-up speed precisely matches the prediction from [2], with sharp asymptotics derived via a refined dynamical decomposition and spectral analysis in a weighted functional framework.

ABSTRACT

We exhibit a stable finite time blow up regime for the 1-corotational energy critical harmonic heat flow from $\\Bbb R^2$ into a smooth compact revolution surface of $\\Bbb R^3$ which reduces to the semilinear parabolic problem $$\\partial_t u -\\pa^2_{r} u-\\frac{\\pa_r u}{r} + \\frac{f(u)}{r^2}=0$$ for a suitable class of functions $f$. The corresponding initial data can be chosen smooth, well localized and arbitrarily close to the ground state harmonic map in the energy critical topology. We give sharp asymptotics on the corresponding singularity formation which occurs through the concentration of a universal bubble of energy at the speed predicted in [Van den Bergh, J.; Hulshof, J.; King, J., Formal asymptotics of bubbling in the harmonic map heat flow, SIAM J. Appl. Math. vol 63, o5. pp 1682-1717]. Our approach lies in the continuation of the study of the 1-equivariant energy critical wave map and Schr\\"odinger map with $\\Bbb S^2$ target in [Rapha\\"el, P.; Rodnianksi, I., Stable blow up dynamics for the critical corotational wave maps and equivariant Yang Mills problems, to appear in Prep. Math. IHES.], [Merle, F.; Rapha\\"el, P.; Rodnianski, I., Blow up dynamics for smooth solutions to the energy critical Schr\\"odinger map, preprint 2011.].

Motivation & Objective

• To establish the existence of stable, finite-time blow-up solutions for the 1-corotational energy-critical harmonic heat flow on $ ^2$ into a smooth revolution surface in $ ^3$.
• To resolve the open problem of sharp asymptotics for type II blow-up in the energy-critical setting, particularly for $k=1$ corotational symmetry.
• To demonstrate that initial data can be smooth, well-localized, and arbitrarily close to the ground state harmonic map in the energy-critical topology while still leading to blow-up.
• To confirm the universal blow-up speed predicted in [2], with precise logarithmic corrections.

Proposed method

• The analysis relies on a dynamical decomposition of the solution into a rescaled harmonic map profile and a remainder, with the blow-up rate parameterized by a scale function $\lambda(t)$.
• A weighted functional framework is constructed to control the remainder term $\varepsilon$, incorporating singular weights at $y=0$ and $y=\infty$ to capture the critical behavior.
• Spectral analysis of the linearized operator $H$ around the harmonic map profile is performed, establishing coercivity estimates in weighted $L^2$ spaces with logarithmic corrections.
• A bootstrap argument is employed to control the evolution of the remainder and the scale function $\lambda(t)$, using interpolation and Hardy-type inequalities.
• The method draws on techniques from prior work on wave and Schr"odinger maps with $\mathbb{S}^2$ target, adapting them to the parabolic setting.
• The proof uses compactness arguments and contradiction to rule out non-trivial kernel components, ensuring the stability of the blow-up profile.

Experimental results

Research questions

• RQ1 Can a stable, finite-time blow-up regime be constructed for the 1-corotational energy-critical harmonic heat flow in the parabolic setting?
• RQ2 What is the precise asymptotic behavior of the blow-up rate $\lambda(t)$, and does it match the prediction from [2]?
• RQ3 Can initial data be chosen smooth, localized, and arbitrarily close to the ground state harmonic map while still leading to blow-up?
• RQ4 Is the blow-up profile universal, independent of initial data within a certain energy-critical class?
• RQ5 What are the sharp weighted estimates on the remainder term that ensure the stability of the blow-up dynamics?

Key findings

• A stable finite-time blow-up solution is constructed for the 1-corotational energy-critical harmonic heat flow, with initial data smooth, localized, and arbitrarily close to the ground state harmonic map in the energy-critical topology.
• The blow-up occurs via concentration of energy in a universal bubble, with the blow-up rate $\lambda(t) \sim \frac{T-t}{|\log(T-t)|^2}$, matching the prediction in [2].
• Sharp asymptotics are derived for the solution profile near the blow-up time, confirming the universality of the blow-up dynamics.
• Coercivity estimates for the linearized operator $H$ are established in a weighted $L^2$ space with logarithmic corrections, essential for controlling the remainder term.
• The remainder term $\varepsilon$ satisfies sharp interpolation and pointwise bounds, including $\|\varepsilon\|_{L^\infty} \lesssim \delta(\alpha^*)$ and $\|A\varepsilon\|_{L^\infty} \lesssim b^2|\log b|^2$, ensuring stability.
• The proof rules out non-trivial kernel components via compactness and contradiction, confirming the uniqueness and stability of the blow-up profile.