Global existence for nonlinear heat equation and harmonic maps between Riemannian manifolds

1. Estimate of higher derivatives.
2. Global existence for nonlinear heat equation.

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Let $ M $ be a compact Riemannian manifold. We want to solve the following nonlinear heat equation where $ F: M \longrightarrow M'\subset B\subset V = \mathbb{R}^N $: \[ \frac{d F_t}{d t} = -\Delta F_t + \Gamma(F_t) (\nabla F_t)^2 \] We have proved that the solution exists in short-time and is smooth whenever it exists. We will now establish long-time existence using continuity method: we will show that if the solution exists on $ [\alpha,\omega_n] $ where $ \omega_n $ is an increasing sequence to $ \omega $, then the solution exists on $ [\alpha,\omega]$. We then apply short-time existence to gain a small open interval where solution still exists. We then conclude that the solution exists globally on $ [\alpha,+\infty) $ since this interval is connected.

The crucial step to prove that the solution can be extended on $ [\alpha,\omega] $ is to uniformly bound all of its derivatives in time of evolution $ [\alpha,\omega) $. These estimates will also be useful to justify the convergence of $ F_t $ in $ C^\infty(M) $ to a smooth function $ F_\infty $ which will eventually be a harmonic map from $ M $ to $ M' $.

Recall that we proved in Corollary cor:bound-2-2, under the hypothesis of negative curvature, the boundedness of $ \|F_t\|_{W^{2,2}(M)} $ by a constant $ C $ depending only on curvatures of $ M, M' $ and the initial total energies. Since $ \frac{dF_t}{dt } $ relates to spatial derivatives of $ F $ by the nonlinear heat equation, it is easy to see that $ \|F_t\|_{W^{2,2}(M\times[\tau,\tau +\delta])} $ is bounded by a constant independent of $ \tau $. We will denote $ W^{k,p}(M\times [\beta,\gamma]) $ by $ W^{k,p}([\beta,\gamma]) $.

Suppose $ \Riem(M') \leq 0 $. There exists a constant $ C $ depending only on $ \delta $, the metrics and initial total energies such that \[ \|F\|_{W^{2,2}(\tau,\tau+\delta)}\leq C\quad\text{for all } \alpha \leq \tau <\omega-\delta. \]

Since \[ \|F\|^2_{W^{2,2}([\tau,\tau+\delta])} \leq \int_\tau^{\tau+\delta} \|F_t\|^2_{W^{2,2}(M)} dt + 2\int_\tau^{\tau+\delta} \left\| \Delta F_t \right\|^2_{L^2}dt + 2\int_\tau^{\tau+\delta} \left\| \Gamma(F_t)(\nabla F_t)^2 \right\|^2_{L^2}dt \] The first term and the second term are bounded by $ C^2\delta $, the third one, since $ \Gamma(F_t) $ is bounded, by $ C^2\delta $ where $ C $ is a constant only depending on the metrics and initial total energies.

The estimates of higher derivatives of $ F $ will be established in the same strategy as the bootstrap: first in $ W^{2,p} $ for all $ p $ then in $ W^{k,p} $ for all $ k,p $, then in $ C^\infty $.

1 Estimate of higher derivatives.

Suppose $ \Riem(M') \leq 0 $. For all $ p \in (1,+\infty) $, there exists a constant $ C>0 $ depending only on $ \delta $, $ p $, the metrics and initial energies such that for all $ \alpha +\delta \leq \tau \leq \omega-\delta $: \[ \|F\|_{W^{2,p}([\tau,\tau+\delta])}\leq C \]

Applying Gårding Inequality to the parabolic equation $ AF = \Gamma(F) (\nabla F)^2 $ where $ A:= \frac{\partial}{\partial t} + \Delta $ is the heat operator, one has \[ \|F\|_{W^{2,p}([\tau,\tau+\delta])} \leq C\left( \|\Gamma(F)(\nabla F)^2\|_{L^p([\tau -\frac{\delta}{3},\tau +\delta ])} + \|F\|_{W^{2,2}([\tau-\frac{\delta}{3},\tau +\delta])} \right) \] The second term of RHS is already bounded by applying Theorem thm:bound-2-2 to $ \frac{4\delta}{3} $. For the first term: \[ \left\| \Gamma(F) (\nabla F)^2\right\|_{L^p([\tau-\frac{\delta}{3},\tau+\delta])} \leq C(M') \| |\nabla F|^2 \|_{L^p([\tau-\frac{\delta}{3}, \tau+\delta])} = C(M') \| e(F) \|_{L^p([\tau - \frac{\delta}{3},\tau+\delta])}. \]

Recall that by Theorem thm:den-pot, the potential density satisfies $ \frac{de}{dt} +\Delta e -Ce \leq 0 $ for certain constant $ C $ depending only on the metric of $ M $. By Maximum principle (Theorem thm:max-princ-d), one has $ e \leq \psi_\tau $ where $ \psi_\tau $ is the solution of

\begin{cases} \frac{d}{dt}\psi_\tau + \Delta\psi_\tau -C\psi_\tau = 0 \\ \restr{\psi_\tau}{\tau - \frac{\delta}{2}} = \restr{e}{\tau - \frac{\delta}{2}} \end{cases}

We apply Gårding Inequality again for $ \psi_\tau $ and obtain

\begin{equation} \label{eq:lem:bound-2-p:2} \| e(F) \|_{L^p([\tau - \frac{\delta}{3},\tau+\delta])} \leq \|\psi_\tau\|_{L^p([\tau- \frac{\delta}{3},\tau+\delta])} \leq C \|\psi_\tau\|_{L^1([\tau- \frac{\delta}{2},\tau+\delta])}. \end{equation}

Now apply $ L^1 $-Comparison Theorem thm:1-comparison-d to $ \psi_\tau $, one has

\begin{equation} \label{eq:lem:bound-2-p:3} \|\psi_\tau\|_{L^1([\tau- \frac{\delta}{2},\tau+\delta])} \leq \int_0^{3\delta/2} \|\restr{\psi_\tau}{\tau-\frac{\delta}{2}}\|_{L^1}e^{Bt}dt \leq \int_0^{3\delta/2}e^{Bt}dt. \|\restr{e}{\tau-\frac{\delta}{2}}\|_{L^1}\leq C. \end{equation}

The lemma follows from \eqref{eq:lem:bound-2-p:2} and \eqref{eq:lem:bound-2-p:3}.

We can now estimate higher order derivatives.

Suppose $ \Riem(M') \leq 0 $. For all $ p\in (1,+\infty) $ and $ k<+\infty $, there exists $ C $ depending only on $ k$, $p$, the metrics and initial energies such that \[ \|F\|_{W^{k,p}([\tau, \tau+\delta])} \leq C \] for all $ \alpha +\delta \leq\tau\leq\omega-\delta $.

Applying Gårding Inequality to the equation $ \frac{dF}{dt} + \Delta F_t = \Gamma(F)(\nabla F)^2 $ then Regularity estimate for the quadratic term (Theorem thm:reg-quad), one has for $ \epsilon \ll \delta $:

\begin{align*} \|F\|_{W^{k,p}([\tau,\tau+\delta])} &\leq C_\epsilon \left( \|F\|_{W^{2,p}([\tau-\epsilon,\tau+\delta])} + \|\Gamma(F)(\nabla F)^2\|_{W^{k-2,p}([\tau-\epsilon,\tau+\delta])}\right) \\ &\leq C_\epsilon\left(1 + C\left(1+\|F\|_{W^{s,q}([\tau-\epsilon,\tau+\delta])}\right)^{q/p}\right) \end{align*}

as long as $ k-1 < s $ and $ \frac{1}{p} > \frac{k}{s}.\frac{1}{q} $. Therefore if $ \|F\|_{W^{s,q}([\tau,\tau+\delta])}\leq C(\delta,s,q) $ for all $ \beta\leq \tau\leq\omega-\delta $ and $ q\in (1,+\infty) $, we just proved that \[ \|F\|_{W^{k,p}([\tau,\tau+\delta])} \leq C({\epsilon}, k,p) \] for all

\begin{cases} \beta + \epsilon \leq \tau \leq \omega-\delta \\ k < s+1, p \in(1,+\infty) \end{cases}

since $ \|F\|_{W^{s,q}([\tau-\epsilon,\tau+\delta])} \leq 2C(\delta,s,q) $.

One can then conclude by induction on $ k $, with step $ \frac{1}{2} $, starting with $ k=2 $ and $ \epsilon=\frac{\delta}{2} $ divided by 2 after each induction step.

2 Global existence for nonlinear heat equation.

Suppose $ \Riem(M') \leq 0 $. The solution of nonlinear heat equation

\begin{equation} \label{eq:thm:global-heat} \frac{dF}{dt} = -\Delta F +\Gamma(F) (\nabla F)^2 \end{equation}

with smooth initial condition exists globally for all time $ t >\alpha $.

Let $ F_n $ be a sequence of solution of \eqref{eq:thm:global-heat} on $ [\alpha,\omega_n] $ with $ \omega_n $ increasing to $ \omega $ then they coincide by uniqueness of the solution. As discussed in the beginning of this part, it is sufficient to prove that the solution extends to $ [\alpha,\omega] $. Let $ F $ be the solution on $ [\alpha,\omega) $ such that $ \restr{F}{[\alpha,\omega_n]} = F_n $, then by Theorem thm:bound-k-p, for all $ \tau \in [\alpha,\omega-\delta) $: \[ \| D^u_t D^v_x F \|_{L^\infty(M\times[\tau,\tau+\delta])} \leq C_{\rm Sobolev} \| D^u_t D^v_x F \|_{W^{k,p}(M\times[\tau,\tau+\delta])} \leq C_{\rm Sobolev} . C(k,p,\delta) \] where, if we choose $ k $ sufficiently large, $ C_{\rm Sobolev} $ is the constant of Sobolev imbedding $ W^{k,p}(M\times[0,\delta]) \hookrightarrow C(M\times[0,\delta]) $ and $ C(k,p,\delta) $ is the constant provided by Theorem thm:bound-k-p.

So all partial derivatives of $ F $ are uniformly bounded on $ [\alpha,\omega) $. This proves that $ F $ extends to a solution on $ [\alpha,\omega] $. In fact $\restr{F}{\tau}:=\restr{F}{M\times\{\tau\}} $ converges in $ C^\infty(M) $ as $ \tau\to \omega $, since \[ \|D^\alpha \restr{F}{\tau} - D^\alpha\restr{F}{\tau'}\|_{L^\infty} \leq \max_{\|\beta\| = \|\alpha\|+1}\|D^{\beta}F\|_{L^\infty}|\tau-\tau'|. \]

We have just proved the first part of the following theorem.

Let $ M, M' $ be compact Riemannian manifolds with $ \Riem(M') \leq 0 $. Then for every smooth map $ f_0:\ M \longrightarrow M'\subset B\subset \mathbb{R}^N $, the nonlinear heat equation
\begin{equation*} \begin{cases} \frac{df_t}{dt} = \tau(f_t), & \text{for all $t\geq 0$} \\ \restr{f}{t=0} = f_0, \end{cases} \end{equation*}
admits a globally defined smooth solution $ f_t $. Moreover, all derivatives $ D^\alpha f_t $ remain bounded as $ t\to +\infty $.
For a suitable sequence $ \{t_n\} $ increasing to $ +\infty $ the sequence $ \{f_{t_n}\} $ converges in $ C^\infty(M) $ to a function $ f_\infty $ with $ \tau(f_\infty)=0 $. Therefore any map $ f_0:\ M \longrightarrow M' $ is homotopic to a harmonic map.

For any sequence $\{t_n\}$, one can extract from $ \{f_{t_n}\} $, since their derivatives are uniformly bounded, a subsequence $ \{f_{t_{n_i}}\} $ converging in $ C^k(M, \mathbb{R}^N) $. By a diagonal argument, one can extract from any sequence $ \{f_{t_n}\} $ a subsequence converging in $ C^\infty(M, \mathbb{R}^N) $ to $ f_\infty $. Abusively denote this subsequence by $ \{f_{t_n}\} $, by Theorem thm:den-kin \[ \lim_{n\to\infty} K(f_{t_n}) = \lim_{n\to\infty} \int_M |\tau(f_{t_n})|^2 = 0 \] Therefore $ \tau(f_{t_n}) \to 0 $ in $ L^2(M)^{\oplus N} $. But also $ \tau(f_{t_n}) \to \tau(f_\infty) $ in $ C^\infty(M, \mathbb{R}^N) $, one has $ \tau(f_\infty)=0 $.

Global existence for nonlinear heat equation and harmonic maps between Riemannian manifolds

Table of Contents

1 Estimate of higher derivatives.

2 Global existence for nonlinear heat equation.