Existence of the heat flow with sign-changing prescribed function. (English) Zbl 1543.58014
Let \((\Sigma, g)\) be a closed Riemann surface, \( u_0 = u(\cdot, 0) \in C^{2+\beta}(\Sigma) \) with \( \beta \in (0,1) \), and \( h \) be a smooth function that changes sign and satisfies \( \int_{\Sigma} h e^{u_0} \, dv_g \neq 0 \).
In this paper, the authors investigate the following mean-field type flow: \[ \left\{ \begin{array}{l} \frac{\partial}{\partial t} e^u + \Delta_g u = \alpha(u - \bar{u}) + 8 \pi \left(\frac{h e^u}{\int_{\Sigma} h e^u \, dv_g} - 1\right), \\ u(\cdot, 0) = u_0, \end{array} \right. \] where \( \bar{u} = \frac{1}{|\Sigma|} \int_{\Sigma} u \, dv_g \), \( 0 \leq \alpha < \lambda_1(\Sigma) \), and \( \lambda_1(\Sigma) \) is the first eigenvalue of the Laplace-Beltrami operator \( \Delta_g \).
This equation can be recognized as a gradient flow of the functional \( J_{8 \pi}(u) \): \[ J_{8 \pi}(u) = \frac{1}{2} \int_{\Sigma} |\nabla_g u|^2 \, dv_g - \frac{\alpha}{2} \int_{\Sigma} (u - \bar{u})^2 \, dv_g + 8 \pi \bar{u} - 8 \pi \log \left|\int_{\Sigma} h e^u \, dv_g\right|. \] The authors establish the following theorem:
{Theorem.} Suppose \( (\Sigma, g) \) is a closed Riemann surface, \( h \in C^{\infty}(\Sigma) \) with \( h \not \equiv 0 \), and \( u_0 \in C^{2+\beta}(\Sigma) \) with \( \beta \in (0,1) \), satisfying \( \int_{\Sigma} h e^{u_0} \, dv_g \neq 0 \). Then, there exists a unique global solution \( u \) to the above mean-field flow equation. Moreover, if \( \Delta_g u = \alpha(u - \bar{u}) + 8 \pi \left(\frac{h e^u}{\int_{\Sigma} h e^u \, dv_g} - 1\right) \) has no solution, then for all \( t \geq 0 \), \[ J_{8 \pi}(u(\cdot, t)) \geq -4 \pi \max_{x \in \Sigma} \left( 2 \log (\pi |h(x)|) + A_x \right) - 8 \pi, \] where \( A_x \) is a constant associated with the Green function \( G_x(y) \), satisfying \[ \left\{ \begin{array}{l} \Delta_g G_x - \alpha G_x = 8 \pi \delta_x - 8 \pi, \\ \int_{\Sigma} G_x \, dv_g = 0. \end{array} \right. \]
More explicitly, the authors show in Lemma 6 that \(G_x\) satisfies \[ G_{x}=-4 \log r+A_{x}+O(r) \text {, where } A_{x} \text { is the constant above. } \] The authors use blow-up analysis to derive their results, and overcome difficulties arising from the term \( \alpha(u - \bar{u}) \).
In this paper, the authors investigate the following mean-field type flow: \[ \left\{ \begin{array}{l} \frac{\partial}{\partial t} e^u + \Delta_g u = \alpha(u - \bar{u}) + 8 \pi \left(\frac{h e^u}{\int_{\Sigma} h e^u \, dv_g} - 1\right), \\ u(\cdot, 0) = u_0, \end{array} \right. \] where \( \bar{u} = \frac{1}{|\Sigma|} \int_{\Sigma} u \, dv_g \), \( 0 \leq \alpha < \lambda_1(\Sigma) \), and \( \lambda_1(\Sigma) \) is the first eigenvalue of the Laplace-Beltrami operator \( \Delta_g \).
This equation can be recognized as a gradient flow of the functional \( J_{8 \pi}(u) \): \[ J_{8 \pi}(u) = \frac{1}{2} \int_{\Sigma} |\nabla_g u|^2 \, dv_g - \frac{\alpha}{2} \int_{\Sigma} (u - \bar{u})^2 \, dv_g + 8 \pi \bar{u} - 8 \pi \log \left|\int_{\Sigma} h e^u \, dv_g\right|. \] The authors establish the following theorem:
{Theorem.} Suppose \( (\Sigma, g) \) is a closed Riemann surface, \( h \in C^{\infty}(\Sigma) \) with \( h \not \equiv 0 \), and \( u_0 \in C^{2+\beta}(\Sigma) \) with \( \beta \in (0,1) \), satisfying \( \int_{\Sigma} h e^{u_0} \, dv_g \neq 0 \). Then, there exists a unique global solution \( u \) to the above mean-field flow equation. Moreover, if \( \Delta_g u = \alpha(u - \bar{u}) + 8 \pi \left(\frac{h e^u}{\int_{\Sigma} h e^u \, dv_g} - 1\right) \) has no solution, then for all \( t \geq 0 \), \[ J_{8 \pi}(u(\cdot, t)) \geq -4 \pi \max_{x \in \Sigma} \left( 2 \log (\pi |h(x)|) + A_x \right) - 8 \pi, \] where \( A_x \) is a constant associated with the Green function \( G_x(y) \), satisfying \[ \left\{ \begin{array}{l} \Delta_g G_x - \alpha G_x = 8 \pi \delta_x - 8 \pi, \\ \int_{\Sigma} G_x \, dv_g = 0. \end{array} \right. \]
More explicitly, the authors show in Lemma 6 that \(G_x\) satisfies \[ G_{x}=-4 \log r+A_{x}+O(r) \text {, where } A_{x} \text { is the constant above. } \] The authors use blow-up analysis to derive their results, and overcome difficulties arising from the term \( \alpha(u - \bar{u}) \).
Reviewer: Junrong Yan (Santa Barbara)
MSC:
| 58J35 | Heat and other parabolic equation methods for PDEs on manifolds |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J60 | Nonlinear elliptic equations |
| 53C20 | Global Riemannian geometry, including pinching |
| 58E15 | Variational problems concerning extremal problems in several variables; Yang-Mills functionals |
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