Heat Flow of Harmonic Mappings

• Heat flow of harmonic mappings is the L²-gradient flow of Dirichlet energy that deforms maps to become harmonic, ensuring optimality in energy reduction.
• The methodology leverages the Caffarelli–Silvestre–Stinga extension to handle fractional operators and employs Ginzburg–Landau penalization to manage boundary constraints.
• Key results demonstrate partial regularity, monotonicity formulas, and asymptotic smoothing, leading to global convergence to stationary states.

The heat flow of harmonic mappings is the evolution equation describing the negative $L^2$-gradient flow of the Dirichlet energy for maps between manifolds, providing a canonical deformation from general maps to harmonic maps (critical points of the energy). The subject encompasses the intricate regularity, compactness, uniqueness, and long-time behavior of weak and strong solutions, and extends to fractional, conformal, and free boundary analogues. This field also connects closely to integrability by compensation, monotonicity formulas, and parabolic partial differential equations with nonlocal operators.

1. Half-Harmonic Maps, Free Boundary Harmonic Maps, and Their Energies

Let $N \subset \mathbb{R}^\ell$ be a compact Riemannian manifold. In one spatial dimension, a half-harmonic map $u: \mathbb{R} \to N$ is a critical point of the $\dot{H}^{1/2}$-energy,

$$E_{1/2}(u) = \frac{1}{2} \int_{\mathbb{R}} |(-\Delta)^{1/4}u(x)|^2\,dx.$$

In the conformal case $m=1$, Da Lio and Rivière demonstrated that critical points satisfy $(-\Delta)^{1/2}u(x) \perp T_{u(x)}N$ distributionally. Millot and Sire established that the harmonic extension $U(x,y)$ of such $u$ from $\mathbb{R}$ into the upper half-plane $\mathbb{R} \times [0,\infty)$ yields a classical harmonic map into $N$ with free boundary, creating a direct correspondence between half-harmonic maps on $\mathbb{R}$ and harmonic maps on the disk with free boundary on $N$. This integrability by compensation is crucial in the one-dimensional setting.

2. The Fractional Heat Flow: Definition and Extension Problem

The half-harmonic heat flow involves the fractional parabolic operator

$$(\partial_t - \Delta)^{1/2} = \mathcal{F}^{-1}_{(x,t)\to(\xi,\sigma)}\left((|\xi|^2 + i\sigma)^{1/2}\right),$$

yielding the flow equation

$$(\partial_t - \Delta)^{1/2}u(x,t) \perp T_{u(x,t)}N, \qquad x \in \mathbb{R}^m,\, t > 0,$$

with $u(x,t) = u_0(x)$ for $t \le 0$. The equation can be recast via the Caffarelli–Silvestre–Stinga extension: introduce $U(x,y,t)$ on $\mathbb{R}^{m+1}_+ \times \mathbb{R}$ solving

$$y^{1-2s}\,\partial_t U = \operatorname{div}_X(y^{1-2s} \nabla_X U)$$

with $U(x,0,t)=u(x,t)$ and boundary condition $$\lim_{y \to 0^+} y^{1-2s} \partial_y U(x,y,t) \perp T_{u(x,t)}N$$. For $s=1/2$, this reduces to a standard heat equation in the half-space paired with a nonlinear Neumann boundary condition, localizing the fractional operator.

3. Weak Formulation and the Ginzburg–Landau Approximation

Weak solutions are constructed in the weighted Sobolev space $H^1(\mathbb{R}_+^{m+1}; y^{1-2s} dX)$. For the pair $(U,u)$, the weak formulation requires:

• $\partial_t U \in L^2_t L^2_X(y^{1-2s})$, $U \in L_{t}^{\infty} H^1_X(y^{1-2s})$,
• $U(x,0,t) = u(x,t)$ for $t \leq 0$,
• for every test field $\Phi$ with $\Phi(x,0,t) \in T_{u(x,t)}N$, the identity

$$\int_0^{\infty} \int_{\mathbb{R}_+^{m+1}} \langle \partial_t U, \Phi \rangle + \langle \nabla_X U, \nabla_X \Phi \rangle\, y^{1-2s} dX dt = 0$$

must hold.

To manage the constraint $u \in N$, a Ginzburg–Landau penalization is introduced on the boundary: $$\lim_{y \to 0^+} y^{1-2s} \partial_y U^\varepsilon = -\frac{c_s}{\varepsilon^2} (1 - |u^\varepsilon|^2) u^\varepsilon.$$ The corresponding penalized energy is

$$\mathscr{E}_\varepsilon(U^\varepsilon) = \frac{1}{2} \int y^{1-2s} |\nabla_X U^\varepsilon|^2 + \frac{c_s}{4\varepsilon^2} \int (1 - |u^\varepsilon|^2)^2,$$

and the time-discretized (Rothe) scheme produces approximate solutions converging to a weak solution of the half-harmonic heat flow.

4. Partial Regularity, Monotonicity, and Energy Quantization

The fundamental existence theorem asserts: for $u_0 \in \dot{H}^{1/2}(\mathbb{R}^m,N)$, there exists a global weak solution $u(x,t)$ with $u(\cdot,t) = u_0$ for $t \leq 0$, and $$u \in L^\infty(\mathbb{R}_+; \dot{H}^{1/2}(\mathbb{R}^m))$$. There is a closed singular set $\Sigma \subset \mathbb{R}^m \times (0,\infty)$ of locally finite parabolic $m$-Hausdorff measure such that $u$ is $C^\infty$ away from $\Sigma$. Each time slice $\Sigma_t$ has finite $(m-1)$-Hausdorff measure.

The monotonicity formula is central: defining

$$\mathcal{D}_\varepsilon(Z_0, R) = R^2 \left[ \frac{1}{2} \int_{\mathbb{R}_+^{m+1}} |\nabla_X U^\varepsilon|^2 G_{Z_0}(X, t_0 - R^2) + \frac{c_s}{4\varepsilon^2} \int_{\mathbb{R}^m} (1 - |u^\varepsilon|^2)^2 G_{Z_0} \right]$$

and analogous $E_\varepsilon(Z_0,R)$, both are nondecreasing in $R$. The local energy inequality and $\varepsilon$-regularity lemma state that sufficiently small energy in a parabolic cylinder implicates regularity in a smaller subcylinder, yielding $C^{1+\alpha}$ bounds.

5. Long-Time Behavior and Asymptotic Smoothing

For sufficiently large $T_0$ (depending on $\|u_0\|_{\dot{H}^{1/2}}$), $$\Sigma \cap [\mathbb{R}^m \times [T_0, \infty)) = \emptyset$$ and

$\|\nabla u(\cdot, t)\|_{L^\infty} \leq C t^{-1/2}.$

As $t \to \infty$, $u(\cdot, t)$ converges in $C^2_{\mathrm{loc}}$ to a constant $p \in N$. This demonstrates global smoothing after finite time and convergence to a stationary (constant) state.

6. Comparison to Classical and Boundary Value Heat Flows

The half-harmonic flow is dual to Struwe's classical harmonic map heat flow with free boundary: in the Struwe program, the interior constraint $u \in N$ is penalized and the boundary condition remains exact, whereas in the half-harmonic setting, the penalization occurs only at the boundary, and the flow is governed in the bulk by a fractional parabolic operator. Both frameworks rely on monotonicity formulas and $\varepsilon$-regularity arguments, but the half-harmonic case employs parabolic extensions and nonlocal operators, exploiting the Caffarelli–Silvestre–Stinga extension to localize the nonlocal evolution.

7. Key Analytical Tools and Implications

Principal formulas and energy identities include:

• Energy dissipation for the penalized flows:

$$\frac{d}{dt} \mathscr{E}_\varepsilon(U^\varepsilon(t)) + \int y^{1-2s} |\partial_t U^\varepsilon|^2 = 0.$$

• Monotonicity in $R$ for $E_\varepsilon(Z_0, R)$ and $\mathcal{D}_\varepsilon(Z_0, R)$.
• $\varepsilon$-regularity: small energy in a parabolic cylinder controls higher regularity.

The framework demonstrates that partial regularity, energy dissipation, and global smoothing arise from a combination of compensatory integrability (in $m=1$), penalization methods adapted to fractional operators, and parabolic $\varepsilon$-regularity.

A plausible implication is that this variational–parabolic–extension method generalizes to fractional harmonic flows in higher dimensions and to more general geometric flows subject to nonlocal constraints, subject to analogous monotonicity and compactness principles.