Parabolic Regularity

6A.5 Weak Solutions and Elliptic Regularity showed that elliptic equations gain two derivatives: if $Lu = f$ and $f$ has $k$ derivatives, $u$ has $k + 2$. Parabolic equations gain two derivatives in space and one in time, which is the same thing measured with the parabolic scaling of 6A.1 What a PDE Is, where one time derivative counts as two space derivatives. This chapter sets up the spaces in which that statement is exact, parabolic Hölder spaces, and states the two estimates that make the gain precise, Schauder and $L^p$. These are the estimates quoted in every short-time existence proof, including Hamilton's and DeTurck's for the Ricci flow.

The chapter then proves, by a method short enough to do completely, a different kind of estimate: Bernstein's gradient bound, which says that a bounded solution of the heat equation has gradient at most $C/\sqrt t$, by applying the maximum principle not to $u$ but to a cleverly chosen combination of $u$ and $|\nabla u|^2$. That method is exactly how Shi's derivative estimates for the Ricci flow are proved, and the chapter's rehearsal is Shi's argument in miniature.

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

• describe weak solutions of parabolic equations and their construction by Galerkin approximation;
• define parabolic cylinders and parabolic Hölder spaces, and explain their scaling;
• state the interior Schauder and $L^p$ estimates and say what each gives;
• state the De Giorgi–Nash–Moser theorem and Moser's Harnack inequality, and say why they matter for nonlinear equations;
• prove $(\partial_t - \Delta)|\nabla u|^2 = -2|\nabla^2u|^2$ and use it to bound $|\nabla u|$ by Bernstein's method.

6.1Cooling rates in a quenched bar

6.2Weak solutions and their regularity

Consider the problem

with $L$ a uniformly elliptic operator as in 6A.5 Weak Solutions and Elliptic Regularity. A weak solution is a function $u \in L^2(0, T; H^1_0(U))$ with $u_t \in L^2(0, T; H^{-1}(U))$ such that, for almost every $t$,

and $u(0) = g$. (Functions of $t$ with values in a Banach space are Bochner integrable functions, the vector-valued version of 3A.3 The Lebesgue Integral; Evans, section 5.9.)

The construction is Galerkin's method: take the first $m$ eigenfunctions $w_1, \dots, w_m$ of the elliptic operator (6A.5 Weak Solutions and Elliptic Regularity), look for $u_m(t) = \sum_{k\leq m}d_k(t)w_k$ satisfying the equation tested against $w_1, \dots, w_m$, which is a linear system of ODEs for $d_k(t)$ (2B.10 Ordinary Differential Equations), and pass to the limit $m \to \infty$ using the energy estimate

which holds uniformly in $m$ and comes from testing the equation with $u_m$ itself, exactly as the energy decay in 6A.3 The Heat Equation on ℝⁿ. Bounded sequences in Hilbert spaces have weakly convergent subsequences (4A.6 Weak Convergence and the Direct Method), and the limit is a weak solution, which is unique (Evans, section 7.1.2).

Regularity then proceeds as in the elliptic case, testing with time and space difference quotients: with smooth data the weak solution is smooth, and, as for the heat equation on $\mathbb{R}^n$, interior smoothness in space holds for every positive time, even with rough initial data (Evans, section 7.1.3). This chapter now turns to estimates in the classical spaces, which are what nonlinear problems need.

6.3Parabolic cylinders and Hölder spaces

Parabolic scaling (6A.1 What a PDE Is) says the right space-time distance is

which is unchanged by $(x, t) \mapsto (\lambda x, \lambda^2t)$ up to the factor $\lambda$. Its balls are, up to constants, the parabolic cylinders

of radius $r$ in space and $r^2$ in time, reaching backward from $t_0$, because a parabolic equation is controlled by its past (Figure 6.1).

The time exponents are half the space exponents, as scaling demands: one time derivative weighs as much as two space derivatives. The heat operator $\partial_t - \Delta$ maps $C^{2+\alpha,1+\alpha/2}$ to $C^{\alpha,\alpha/2}$, losing exactly two units of parabolic regularity, and the Schauder estimate says it loses no more.

6.4The Schauder and $L^p$ estimates

Consider the non-divergence operator

uniformly parabolic: $\sum a^{ij}\xi_i\xi_j \geq \theta|\xi|^2$.

The proof (Lieberman, chapter 4; Krylov) freezes the coefficients at a point, compares with the constant-coefficient equation, which is the heat equation after a linear change of variables and can be estimated using the heat kernel, and controls the error by the Hölder continuity of the coefficients on small cylinders. There are versions up to the boundary and for the initial time, with compatibility conditions on the data. The form to remember is: data in $C^\alpha$, solution in $C^{2+\alpha}$, with a scaled estimate on every cylinder $Q_r$ obtained by applying the theorem to $u(x_0 + rx, t_0 + r^2t)$ (Exercise 6.6). Written out for $u_t = \Delta u + f$, it reads $r^{2+\alpha}[\nabla^2u]_{\alpha;Q_{r/2}} \leq C\big(\sup_{Q_r}|u| + r^2\sup_{Q_r}|f| + r^{2+\alpha}[f]_{\alpha;Q_r}\big)$: every term has the units of $u$.

The $L^p$ estimates are the other standard form: under continuity of $a^{ij}$,

They come from the theory of singular integrals of Calderón and Zygmund, and fail at $p = 1$ and $p = \infty$ for the same reason as $C^2$ fails (4A.11 Hölder Spaces). $L^p$ estimates with large $p$, combined with the Sobolev embedding (4A.10 Sobolev Embeddings and Critical Exponents), give Hölder continuity of $\nabla u$, which is often the first step of a bootstrap.

Both estimates need some continuity of the leading coefficients. For nonlinear equations that is a problem: the coefficients depend on the solution, whose continuity is what one is trying to prove. The breakthrough that resolves it is the De Giorgi–Nash–Moser theorem: for divergence-form equations $u_t = \partial_j(a^{ij}(x, t)\partial_iu)$ with merely bounded measurable, uniformly elliptic coefficients, solutions are Hölder continuous, with estimates depending only on the ellipticity constants (De Giorgi 1957 for elliptic equations, Nash 1958 for parabolic ones, Moser's simplified proofs 1960–64). Its companion is Moser's parabolic Harnack inequality (1964): a positive solution on $Q_{2r}$ satisfies

where $Q^-$ is a cylinder in the earlier part of $Q_{2r}$ and $Q^+$ one in the later part, separated by a time gap: positive solutions cannot vary wildly, but heat must have time to arrive (6A.2 Harmonic Functions, 6A.10 Entropy, Information and Diffusion). Nash's proof used an entropy, which is the subject of 6A.10 Entropy, Information and Diffusion.

6.5Bernstein's method

All of the above is quoted in Ricci flow papers. The next argument is used: its structure is the structure of Shi's derivative estimates, and of many curvature estimates after them.

Both quantities are subsolutions: $|\nabla u|^2$ and $u^2$ can only decrease along the flow, in the sense of the maximum principle. The good negative term $-2|\nabla u|^2$ in the evolution of $u^2$ is what lets one control $|\nabla u|^2$:

Compare the exact bound $|\nabla u| \leq \frac{M}{\sqrt{\pi t}}$ from the heat kernel (6A.3 The Heat Equation on ℝⁿ, exercises): Bernstein's constant is slightly worse, but the method uses no formula for the solution, only the equation and the maximum principle. So it works on manifolds, for variable coefficients, and for nonlinear equations, where no kernel is available. The weight $t$ handles the initial time, where no gradient bound is assumed; the term $\frac12u^2$ supplies the negative $|\nabla u|^2$ that cancels the derivative of the weight. Higher derivatives are handled the same way, by induction, with $F_k = t^k|\nabla^ku|^2 + c_kt^{k-1}|\nabla^{k-1}u|^2 + \dots$ (Exercise 6.9).

6.6History

Juliusz Schauder proved his estimates for elliptic equations in 1934; the parabolic theory was developed in the following decades and assembled in the 1967 monograph of Olga Ladyzhenskaya, Vsevolod Solonnikov and Nina Ural'tseva. Alberto Calderón and Antoni Zygmund's singular integrals date from 1952. Ennio De Giorgi (1957) and John Nash (1958) independently proved Hölder continuity for equations with bounded measurable coefficients, solving Hilbert's nineteenth problem; Jürgen Moser gave new proofs and the Harnack inequalities (elliptic 1961, parabolic 1964). Sergei Bernstein introduced his method of gradient bounds at the beginning of the twentieth century in his work on nonlinear elliptic equations. Shi's derivative estimates appeared in 1989. The Jominy test dates from 1938.

6.7Exercises

Solution

$\partial_t\operatorname{erf}(x/2\sqrt{\kappa t}) = \frac{2}{\sqrt\pi}e^{-x^2/4\kappa t}\cdot\big(-\frac{x}{4\sqrt\kappa}t^{-3/2}\big)$, giving the formula. Maximise $t^{-3/2}e^{-a/t}$ with $a = \frac{x^2}{4\kappa}$: the derivative of $-\frac32\log t - \frac at$ is zero at $t = \frac{2a}{3} = \frac{x^2}{6\kappa}$. Substituting, $t^{-3/2} = (6\kappa)^{3/2}x^{-3}$ and $e^{-a/t} = e^{-3/2}$, so the maximum is $\frac{T_0x}{2\sqrt{\pi\kappa}}(6\kappa)^{3/2}x^{-3}e^{-3/2}$, as stated. At $3$ cm it is $\frac19$ of that at $1$ cm.

Solution

(a) $v_t = \Delta v + r^2f(rx, r^2t)$. (b) Chain rule; Hölder quotients pick up $r^\alpha$ from $|x - y|^\alpha$. (c) $r^2\sup_{Q_{r/2}}|\nabla^2u| + r^{2+\alpha}[\nabla^2u]_{\alpha;Q_{r/2}} \leq C\big(\sup_{Q_r}|u| + r^2\sup_{Q_r}|f| + r^{2+\alpha}[f]_{\alpha;Q_r}\big)$.

Solution

$\partial_t(vw) = v_tw + vw_t$ and $\Delta(vw) = v\Delta w + w\Delta v + 2\nabla v\cdot\nabla w$. For $\phi(u)$: $\partial_t\phi(u) = \phi'u_t$ and $\Delta\phi(u) = \phi'\Delta u + \phi''|\nabla u|^2$, so $(\partial_t - \Delta)\phi(u) = -\phi''(u)|\nabla u|^2 \leq 0$ for convex $\phi$. The case $\phi(u) = u^2$ is in Proposition 6.3.

Solution

(a) As in Proposition 6.3, with $v = w = \partial_i\partial_ju$. (b) $(\partial_t - \Delta)G = 2t|\nabla^2u|^2 - 2t^2|\nabla^3u|^2 + A|\nabla u|^2 - 2At|\nabla^2u|^2 - 2B|\nabla u|^2$. With $A = 1$ and $B = \frac12$ every term is $\leq 0$. So $G \leq \sup G(0) = BM^2$, and $t^2|\nabla^2u|^2 \leq \frac12M^2$.

Solution

$(\partial_t - \Delta)F \leq |\nabla T|^2 + Ct K|\nabla T|^2 - 2\beta|\nabla T|^2 + C\beta K^3$. Since $tK \leq 1$, choosing $\beta \geq \frac{1 + C}{2}$ makes the coefficient of $|\nabla T|^2$ non-positive, so $(\partial_t - \Delta)F \leq C\beta K^3$. By the maximum principle $F \leq \sup F(0) + C\beta K^3t \leq \beta K^2 + C\beta K^2 = C''K^2$, using $t \leq 1/K$. Hence $t|\nabla T|^2 \leq C''K^2$.