Quasilinear Parabolic Systems

6C.11 Optimal Transport and Ricci Curvature ended with the Ricci flow characterised by how it moves probability measures. To use the Ricci flow at all one must first solve it, and that is a question about systems. The equations of geometric analysis are mostly systems: the unknown is a map (harmonic map flow), a metric (Ricci flow), an immersion (mean curvature flow), with several components coupled through their second derivatives. Scalar intuition is a dangerous guide to systems. The maximum principle, the backbone of Books 6A and 6B, fails for general systems (Exercise 12.8); what survives is the theory based on Fourier analysis and energy, organised around the principal symbol.

This chapter does three things. It defines parabolicity for systems through the symbol and shows why that is the right condition. It proves short-time existence and uniqueness for quasilinear parabolic systems, by linearising and applying the contraction mapping principle of 6A.8 Nonlinear Parabolic Equations, using the linear Schauder theory as a black box. And it computes the symbol of the Ricci flow, finds the $n$-dimensional kernel forced by diffeomorphism invariance, and shows how DeTurck's modification removes it, so that the existence theorem applies. This is the analytic foundation of 11A.4 Short-Time Existence and Uniqueness.

By the end of this chapter you will be able to:

• compute the principal symbol of a second-order system and decide whether it is strongly parabolic;
• solve constant-coefficient parabolic systems by Fourier transform, and show that a degenerate or negative symbol destroys smoothing or well-posedness;
• prove that a function vanishing at $t = 0$ is small in a lower parabolic Hölder norm on a short time interval, and use it to prove short-time existence and uniqueness by contraction;
• compute the symbol of $-2\operatorname{Ric}$, its kernel, and its eigenvalues;
• show that the Ricci–DeTurck flow is strongly parabolic, with symbol $|\xi|^2$ times the identity.

12.1Stripes and spots

12.2The principal symbol

Let $u : \mathbb{R}^n\times[0, T) \to \mathbb{R}^N$ (or $u$ a section of a vector bundle over a manifold), and consider a second-order quasilinear system

with summation over $i, j$ and $b$. The principal symbol at $(x, u, \nabla u)$ in the direction $\xi \in \mathbb{R}^n$ is the $N\times N$ matrix obtained by replacing each $\partial_i$ in the top-order part by $\xi_i$:

For a scalar equation ($N = 1$) this is ordinary uniform parabolicity. For diagonal systems $a^{ij}_{ab} = a^{ij}\delta_{ab}$, such as the harmonic map heat flow ($\sigma(\xi) = |\xi|^2I$, 6C.7 Harmonic Maps and Their Heat Flow) or the Ricci–DeTurck flow below, it is the same condition for each component. In general the condition involves only rank-one tensors $\xi\otimes\eta$, which is weaker than positivity of $a$ on all of $\mathbb{R}^{nN}$ (the Legendre condition).

Why the symbol? Freeze the coefficients at a point and drop lower-order terms: $\partial_tu = a^{ij}\partial_{ij}u$ with constant matrices $a^{ij}$. Fourier transform in $x$ turns it into the ODE system $\partial_t\hat u(\xi, t) = -\sigma(\xi)\hat u(\xi, t)$, so $\hat u(\xi, t) = e^{-t\sigma(\xi)}\hat u_0(\xi)$.

Part 2 is exactly the situation of the Ricci flow: it has stationary-looking directions in which no smoothing occurs, the directions of infinitesimal diffeomorphisms. A rough diffeomorphism applied to a smooth metric stays rough, and no amount of Ricci flow can smooth it, because the flow is diffeomorphism invariant.

12.3Linear theory (statement)

We use parabolic Hölder spaces on a closed manifold $M$ (or a torus) and a time interval $[0, T]$, $T \leq 1$, as in 6A.7 Parabolic Regularity: $C^{\alpha,\alpha/2}$, $C^{2+\alpha,1+\alpha/2}$, and the intermediate space $C^{1+\alpha,(1+\alpha)/2}$ of functions $u$ such that $u$ and $\nabla u$ are continuous, $\nabla u \in C^{\alpha,\alpha/2}$, and $u$ is $\frac{1+\alpha}{2}$-Hölder in $t$. Write $\|\cdot\|_{k+\alpha}$ for the norms. For vector-valued functions these are taken componentwise.

For scalar equations this is 6A.7 Parabolic Regularity together with the existence theory of 6B.11 Linear Parabolic Equations in Full; for systems it is due to Vsevolod Solonnikov (1965), and the proof is the same in outline: freeze coefficients, solve the constant-coefficient problem by an explicit kernel (here $e^{-t\sigma(\xi)}$ in Fourier variables), prove Hölder estimates for it, and patch by the continuity method (6B.7 Schauder Theory and the Continuity Method). The maximum principle is never used, which is why the theory extends from equations to systems.

12.4Short-time existence

The key observation is elementary: a function that vanishes at $t = 0$ and has a bound in a strong norm is small in a weaker norm for a short time.

We now prove existence for quasilinear systems whose leading coefficients depend on $u$ but not on $\nabla u$:

This class contains the harmonic map heat flow and the Ricci–DeTurck flow, the two systems needed for Ricci flow. (When $a^{ij}$ depends on $\nabla u$ too, the same strategy works with the contraction carried out in $C^{2+\alpha}$-type spaces, or with the inverse function theorem in Banach spaces as Hamilton did; Taylor III, §15.7.)

Regularity and continuation. If the data are smooth, so is the solution for $t > 0$ and up to $t = 0$ if $u_0$ is smooth: differentiate the equation in $x$, and apply Theorem 12.3 to the derivatives, which satisfy linear systems of the same type. And the solution continues as long as its $C^{1+\alpha}$ norm stays bounded and the system stays parabolic along it, because the existence time in the proof depends only on such bounds. For the Ricci flow, Shi's estimates turn a bound on curvature into bounds on all derivatives, so the flow continues as long as the curvature is bounded (11A.4 Short-Time Existence and Uniqueness).

12.5The symbol of the Ricci flow

To apply this theorem to $\partial_tg = -2\operatorname{Ric}(g)$, we need the symbol of the linearisation. Since only top-order terms matter, compute at a point in coordinates where $g_{ij} = \delta_{ij}$, and keep only the second derivatives of the variation $h = \delta g$. From $\Gamma^k_{ij} = \frac12g^{kl}(\partial_ig_{jl} + \partial_jg_{il} - \partial_lg_{ij})$ and $R_{ij} = \partial_k\Gamma^k_{ij} - \partial_j\Gamma^k_{ik} + (\Gamma\Gamma\text{ terms})$,

(11A.2 How Curvature Evolves, Exercise 12.10). Replacing $\partial_k$ by $\xi_k$, the symbol of $-2\delta\operatorname{Ric}$ is

The kernel consists of the symbols of Lie derivatives: $\mathcal L_Xg$ has components $\nabla_iX_j + \nabla_jX_i$, whose symbol is $\xi\otimes X + X\otimes\xi$. This is no accident. Since $\operatorname{Ric}(\phi^*g) = \phi^*\operatorname{Ric}(g)$ for every diffeomorphism $\phi$, differentiating along the flow $\phi_s$ of a vector field $X$ gives $\delta\operatorname{Ric}(\mathcal L_Xg) = \mathcal L_X\operatorname{Ric}(g)$, which involves only first derivatives of $X$. A second-order operator with non-degenerate symbol, applied to $\mathcal L_Xg$ (which is first order in $X$), would produce third derivatives of $X$; so the symbol must vanish on $\xi\otimes X + X\otimes\xi$. Diffeomorphism invariance forces a degenerate symbol, by Proposition 12.2 part 2 there is no smoothing in those directions, and Theorem 12.5 does not apply.

Hamilton's original proof (1982) handled the degeneracy with the Nash–Moser implicit function theorem, using the fact that the degeneracy is exactly accounted for by the Bianchi identity (6C.3 Symmetry, Noether’s Theorem and Moving Planes, Exercise 12.13). DeTurck found a much simpler way a year later.

12.6DeTurck's trick

Fix a background metric $\tilde g$ with Christoffel symbols $\tilde\Gamma$, and let

a vector field (a difference of connections is a tensor), equal to minus the tension field of the identity map $(M, g) \to (M, \tilde g)$ (6C.7 Harmonic Maps and Their Heat Flow). The Ricci–DeTurck flow is

So Theorem 12.5 applies to (RD): on a closed manifold, for any smooth initial metric there is a unique smooth solution of the Ricci–DeTurck flow for a short time. To return to the Ricci flow, solve the ODE $\partial_t\phi_t = -W(\phi_t, t)$, $\phi_0 = \mathrm{id}$, for a family of diffeomorphisms; then $\phi_t^*$ of the Ricci–DeTurck solution solves the Ricci flow with the same initial metric (the Lie derivative term is exactly undone by pulling back along the flow of $W$). This proves short-time existence. For uniqueness, Hamilton observed (1995) that, conversely, given a Ricci flow, the diffeomorphisms that convert it into a Ricci–DeTurck flow solve the harmonic map heat flow from $(M, g(t))$ to $(M, \tilde g)$, a strongly parabolic system to which the same theorem applies; uniqueness of both systems then gives uniqueness of the Ricci flow. The details are in 11A.4 Short-Time Existence and Uniqueness.

12.7History

Ivan Petrovskii defined parabolic systems in 1938, and Samuil Eidelman's book on parabolic systems appeared in 1964. Vsevolod Solonnikov proved the Schauder estimates for general parabolic systems in 1965, and Ladyzhenskaya, Solonnikov and Ural'tseva's book (1967 in Russian, 1968 in English) became the standard reference for quasilinear parabolic theory. Herbert Amann developed a general theory of quasilinear parabolic systems in the 1980s. Alan Turing's reaction–diffusion paper appeared in 1952. Hamilton proved short-time existence for the Ricci flow in 1982 with the Nash–Moser theorem; Dennis DeTurck's simplification appeared in 1983, and Hamilton's harmonic-map approach to uniqueness in 1995.

12.8Exercises

Solution

(a) $\operatorname{tr}A = -\frac12 < 0$, $\det A = -\frac32 + 2 = \frac12 > 0$, so both eigenvalues have negative real part. (b) The mode solves $\eta' = (A - k^2D)\eta$. The trace $-\frac12 - k^2(1 + d)$ is always negative, so an eigenvalue with positive real part exists iff the determinant is negative (a real $2\times2$ matrix with negative trace has an eigenvalue with positive real part iff its determinant is negative). (c) With $s = k^2$: $\det(A - sD) = (1 - s)(-\frac32 - ds) + 2 = ds^2 + (\frac32 - d)s + \frac12$. Its minimum over $s > 0$ (for $d > \frac32$) is at $s = \frac{d - 3/2}{2d}$, with value $\frac12 - \frac{(d - 3/2)^2}{4d}$, negative iff $(d - \frac32)^2 > 2d$, i.e. $d^2 - 5d + \frac94 > 0$, i.e. $d > \frac92$ (the other root $\frac12$ is below $\frac32$). At $d = 8$: $s = \frac{6.5}{16} = 0.40625$, $k \approx 0.637$, wavelength $2\pi/k \approx 9.9$. Diffusion alone, applied to a stable system, created instability at a preferred wavelength.

Solution

For $P = \xi\otimes\eta$, $P_{ij} = \xi_i\eta_j$ and $\det P = \xi_1\eta_1\xi_2\eta_2 - \xi_1\eta_2\xi_2\eta_1 = 0$, and $|P|^2 = |\xi|^2|\eta|^2$. For $P = I$: $Q = 2 + \kappa$, negative if $\kappa < -2$; for $P = \operatorname{diag}(1, -1)$: $Q = 2 - \kappa$, negative if $\kappa > 2$. (The determinant is a null Lagrangian, 6C.5 Fixed Points, Monotonicity and Sub- and Supersolutions: adding it changes the energy form but not the Euler–Lagrange equations, so it cannot affect parabolicity, which depends only on the equation.)

Solution

$\delta\Gamma^k_{ij} = \frac12g^{kl}(\partial_ih_{jl} + \partial_jh_{il} - \partial_lh_{ij}) + (\delta g^{kl})(\ldots)$, and the second term involves $\partial g = 0$. Then $\partial_k\delta\Gamma^k_{ij} = \frac12(\partial_k\partial_ih_{jk} + \partial_k\partial_jh_{ik} - \partial_k\partial_kh_{ij})$ and $\partial_j\delta\Gamma^k_{ik} = \frac12(\partial_j\partial_ih_{kk} + \partial_j\partial_kh_{ik} - \partial_j\partial_kh_{ik}) = \frac12\partial_i\partial_jH$. The difference is the stated formula; the $\Gamma\Gamma$ terms contribute only first derivatives of $h$.

Solution

$\xi\lrcorner h = (h_{11}, h_{12}, \ldots, h_{1n})$. $(\sigma h)_{11} = h_{11} + H - 2h_{11} = H - h_{11} = \sum_{\alpha\geq2}h_{\alpha\alpha}$. $(\sigma h)_{1\alpha} = h_{1\alpha} + 0 - h_{1\alpha} - 0 = 0$. $(\sigma h)_{\alpha\beta} = h_{\alpha\beta}$. In the ordered basis ($h_{\alpha\beta}$ block, then $h_{11}, h_{1\alpha}$), $\sigma$ is block triangular with the identity on the first block and zero on the second, so its eigenvalues are $1$ (multiplicity $\frac{n(n-1)}{2}$) and $0$ (multiplicity $n$). It is not diagonalisable: $h = e_2\otimes e_2$ maps to itself plus $e_1\otimes e_1$.

Solution

$H = nf$ and $\xi\lrcorner h = f\xi$, so $\sigma h = |\xi|^2fg + nf\xi\otimes\xi - 2f\xi\otimes\xi$. For $n = 2$, $\sigma h = f|\xi|^2g$. On a surface, writing the metric as $e^{2w}g_{\text{flat}}$ in isothermal coordinates already fixes the diffeomorphism freedom (up to conformal maps, which form a finite-dimensional or rigid family on compact surfaces), so the flow of the conformal factor is a scalar strictly parabolic equation. The degenerate directions of the symbol are the Lie derivative directions, which have been removed from the ansatz.

Solution

(a) $\nabla^iR_{ij} = \frac12\nabla_jR$ is $\operatorname{div}\operatorname{Ric} - \frac12\nabla\operatorname{tr}\operatorname{Ric} = 0$. (b) From the proof, $\delta W_k = \partial_ih_{ik} - \frac12\partial_kH + \text{lower order}$, the top-order part of $B(h)_k$. (c) By Noether's second theorem (6C.3 Symmetry, Noether’s Theorem and Moving Planes, Proposition 12.6), invariance of the total scalar curvature under diffeomorphisms gives an identity satisfied by its gradient, $B$ applied to the Einstein tensor vanishes; the symbol of $\operatorname{Ric}$ in the gauge directions is correspondingly degenerate. DeTurck adds $\mathcal L_Wg$, where $W$ is built from $B$, so that the gauge directions acquire exactly the missing symbol ($\xi\otimes(\xi\lrcorner h) + \ldots$), making the total symbol $|\xi|^2$; the cost is a time-dependent diffeomorphism, which is undone afterwards.