[Paper Review] Existence of bubbling solutions without mass concentration

Youngae Lee, Chang‐Shou Lin+1 more|arXiv (Cornell University)|Jul 12, 2018

Advanced Mathematical Physics Problems30 references3 citations

TL;DR

This paper constructs the first explicit example of a non-concentrated bubbling solution to the mean field equation with collapsing singularities, demonstrating that mass concentration does not necessarily occur when vortex points coalesce. Using asymptotic analysis and a Lyapunov-Schmidt reduction method, the authors prove the existence of solutions where blow-up occurs without local mass accumulation, challenging the classical 'bubbling implies mass concentration' principle under vortex collapse.

ABSTRACT

The seminal work \cite{bm} by Brezis and Merle has been pioneering in studying the bubbling phenomena of the mean field equation with singular sources. When the vortex points are not collapsing, the mean field equation possesses the property of the so-called "bubbling implies mass concentration". Recently, Lin and Tarantello in \cite{lt} pointed out that the "bubbling implies mass concentration" phenomena might not hold in general if the collapse of singularities occurs. In this paper, we shall construct the first concrete example of non-concentrated bubbling solution of the mean field equation with collapsing singularities.

Motivation & Objective

• To challenge the classical 'bubbling implies mass concentration' principle in mean field equations when vortex points collapse.
• To construct a concrete example of a bubbling solution that does not concentrate mass despite blow-up behavior.
• To extend the understanding of blow-up phenomena in singular mean field equations beyond the classical framework.
• To provide a rigorous existence proof for non-concentrated bubbling solutions under vortex collapse via asymptotic analysis.

Proposed method

• Employing a Lyapunov-Schmidt reduction method to decompose the solution into a singular part and a regular part.
• Using a modified Green's function $ G_t^{(2)}(x) = 4 au G(x,t\vec{e}) + 4 au G(x,-t\vec{e}) $ to model the interaction of two collapsing vortex points.
• Defining a transformed equation (1.4) with a weight function $ h(x) $ that incorporates the remaining singular sources and curvature effects.
• Applying a cut-off function $ \overline{\chi_{t,q}}(z) $ and scaling to localize the analysis near the collapse point $ \mathfrak{q} $.
• Using integration by parts and estimates on the error terms to control the nonlinear perturbation and solve for the correction $ \phi_{t,q} $.
• Establishing the existence of a solution by solving $ c_{t,q,j} = 0 $ for $ j=1,2 $, ensuring the solution is orthogonal to the kernel of the linearized operator.

Experimental results

Research questions

• RQ1 Can bubbling solutions exist without mass concentration when vortex points collapse?
• RQ2 Does the classical 'bubbling implies mass concentration' principle fail under vortex collapse?
• RQ3 What conditions allow for non-concentrated blow-up in mean field equations with singular sources?
• RQ4 How does the interaction of collapsing vortices affect the asymptotic behavior of solutions?

Key findings

• The paper constructs the first explicit example of a non-concentrated bubbling solution in the mean field equation with collapsing singularities.
• It proves that mass concentration does not occur even when solutions blow up, contradicting the classical 'bubbling implies mass concentration' principle.
• A solution exists where $ \overline{u}_t \to \overline{w} $ uniformly in $ C^2(M \setminus \{\mathfrak{q}\}) $ as $ t \to 0 $, with no local mass accumulation at the collapse point $ \mathfrak{q} $.
• The solution is obtained via a Lyapunov-Schmidt reduction, and the existence of a solution is shown by solving $ c_{t,q,j} = 0 $ for $ j=1,2 $, ensuring orthogonality to the kernel.
• The method confirms that for $ \alpha_1 = \alpha_2 = 1 $, $ \rho \in (8\pi, 16\pi) $, and collapsing vortices, non-concentrated blow-up is possible.
• The analysis shows $ \nabla H_{t,q}(q) $ is $ O(t) $, and the existence of $ q_t = O(t) $ satisfying the orthogonality condition ensures the solution exists without mass concentration.