Linear Parabolic Equations in Full

6B.10 Harmonic Coordinates and the Regularity of Metrics ended with the Ricci–DeTurck flow, a parabolic system with diagonal principal part, and with the remark that its theory is the theory of linear parabolic equations applied component by component. 6A.7 Parabolic Regularity built weak solutions of parabolic equations by Galerkin's method and stated, without proof, two facts it needed: that such a solution is continuous in time with values in $L^2$, and that its $L^2$ norm is differentiable with the obvious formula. It also stated that solutions are as smooth as the data allow. This chapter proves these facts and develops the rest of the Hilbert-space theory of linear parabolic equations, Evans's section 7.1 in full: improved regularity by testing with the time derivative, higher regularity by differentiating in time, instant smoothing by weighted energies, and finally the compactness lemma of Aubin and Lions, which is what makes the Galerkin method work for nonlinear equations.

Two points of view run side by side. The energy method works with integrals: test the equation with the solution, or its derivative, or a weighted version, integrate by parts, and use Gronwall's inequality. The spectral method expands in eigenfunctions, which gives explicit formulas when the operator does not depend on time. The energy method is more general and is the one that extends to nonlinear and geometric problems; the spectral method is the one to compute with. The chapter's anchor, drug release from a polymer film, uses both.

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

• work with Banach-space-valued functions of time, and prove that $u \in L^2(0, T; H^1_0)$ with $u' \in L^2(0, T; H^{-1})$ is continuous into $L^2$ with $\frac{d}{dt}\|u\|^2 = 2\langle u', u\rangle$;
• prove improved regularity, $u \in L^2(0, T; H^2)\cap L^\infty(0, T; H^1_0)$ with $u' \in L^2(0, T; L^2)$, for $H^1_0$ data;
• prove instant smoothing for $L^2$ data by weighted energy estimates, and explain the compatibility conditions needed for smoothness up to $t = 0$;
• prove Ehrling's lemma and the Aubin–Lions compactness lemma, and use them to solve a semilinear equation by Galerkin's method;
• compute short-time and long-time behaviour of diffusion out of a slab, and explain the $\sqrt t$ law.

11.1Drug release from a polymer film

11.2Functions of time with values in a Banach space

A solution of a parabolic equation is, at each time, a function of space: $t \mapsto u(t) \in X$, where $X$ is a space such as $L^2(U)$ or $H^1_0(U)$. For a Banach space $X$, $L^p(0, T; X)$ is the space of (strongly measurable) functions $u : [0, T] \to X$ with

with the usual modification for $p = \infty$; $C([0, T]; X)$ is the space of continuous $u$ with the sup norm. Integrals of $X$-valued functions are Bochner integrals, defined as limits of integrals of simple functions exactly as in 3A.3 The Lebesgue Integral; they satisfy $\|\int u\| \leq \int\|u\|$ and commute with bounded linear maps (Evans, appendix E.5). A function $v \in L^1(0, T; X)$ is the weak derivative $u'$ of $u \in L^1(0, T; X)$ if $\int_0^T\phi'(t)u(t)\,dt = -\int_0^T\phi(t)v(t)\,dt$ for every scalar test function $\phi \in C_c^\infty(0, T)$, the definition of 4A.9 Distributions and Weak Derivatives with vector values.

Let $U$ be bounded, $H^1_0 = H^1_0(U)$, $L^2 = L^2(U)$ and $H^{-1}$ the dual of $H^1_0$, with the pairing $\langle\cdot, \cdot\rangle$; via $f \mapsto (v \mapsto \int fv)$, $L^2 \subset H^{-1}$, and

each inclusion continuous with dense image. A weak solution of a parabolic equation lies in $L^2(0, T; H^1_0)$ with $u' \in L^2(0, T; H^{-1})$ (6A.7 Parabolic Regularity). The next theorem says such a function is better than it looks.

The theorem is the rigorous form of "multiply by $u$ and integrate", the first move of every energy estimate, for functions that are not smooth. Its key structural point is the Gelfand triple $H^1_0 \subset L^2 \subset H^{-1}$: the time derivative lives in a big space, the solution in a small one, and their pairing makes sense in the middle.

11.3Improved regularity

For the rest of the chapter, let

be symmetric and uniformly elliptic with smooth coefficients independent of $t$, $c \geq 0$, and $B[u, v]$ its bilinear form, so that $B[u, u] \geq \theta\|u\|_{H^1_0}^2$ (6A.6 Weak Solutions and Elliptic Regularity). The problem is

with $\partial U$ smooth. 6A.7 Parabolic Regularity constructed the weak solution as a limit of Galerkin approximations $u_m$ in the span of the first $m$ eigenfunctions $w_1, \ldots, w_m$ of $L$ (for a symmetric $L$ we may use its own eigenfunctions, orthonormal in $L^2$ and orthogonal for $B$).

The idea is the parabolic counterpart of the elliptic bootstrap: one more time derivative tested gives one more space derivative, through the elliptic equation $Lu = f - u'$ at each time. Repeating, with time derivatives of the equation:

The compatibility conditions say that the time derivatives of $u$ at $t = 0$, which the equation computes from the data ($g_0 = g$, $g_1 = f(0) - Lg_0$, $g_2 = f'(0) - Lg_1$, and so on), must vanish on $\partial U$, because $u = 0$ there for all time. For the heat equation with $f = 0$, the first condition is $g = 0$ on $\partial U$, the next is $\Delta g = 0$ on $\partial U$, and so on. If they fail, the solution is still smooth for $t > 0$ but not up to the corner $\partial U\times\{0\}$: for the drug-release problem, where the initial concentration is $1$ but the boundary value is $0$, the solution has a singularity at the corners at $t = 0$, which is exactly what produces the $\sqrt t$ law.

11.4Instant smoothing for rough data

Away from $t = 0$, no compatibility is needed: parabolic equations smooth rough data instantly. The weighted energy method proves it with nothing but the equation, in the spirit of Bernstein's estimate (6A.7 Parabolic Regularity), but with integrals instead of maxima.

Each step is the previous one applied to a time derivative, with one more power of $t$ as a weight: $t^kB[\partial_t^{k-1}u, \partial_t^{k-1}u]$ and $t^{k+1}\|\partial_t^ku\|^2$ are bounded by $C_k\|g\|^2$. Combined with elliptic regularity at each time ($L\partial_t^ku = -\partial_t^{k+1}u$), this gives $u(t) \in H^{2k}(U)$ for every $k$ and every $t > 0$, with $\|u(t)\|_{H^{2k}} \leq C_kt^{-k}\|g\|_{L^2}$: the solution is $C^\infty$ in $x$ for $t > 0$, with the rate $t^{-1/2}$ per derivative that parabolic scaling demands (6A.3 The Heat Equation on ℝⁿ). For the time derivatives the same holds, so $u \in C^\infty(\overline U\times(0, T])$.

Maximum principles and Harnack. The pointwise theory of 6A.5 Maximum Principles applies to classical solutions of $u_t + Lu = 0$ with smooth coefficients, and by the regularity just proved, to every weak solution at positive times: the weak and strong maximum principles hold, and a non-negative solution is either identically zero or positive everywhere at later times. For smooth coefficients, Evans (section 7.1.4) proves the parabolic Harnack inequality by a Li–Yau type estimate for $\log u$; for measurable coefficients it is Moser's theorem (6B.9 De Giorgi–Nash–Moser).

11.5Compactness in time: the Aubin–Lions lemma

Galerkin's method for a linear equation needs only weak convergence: the equation is linear in $u$, so weak limits of approximate solutions are solutions. For a nonlinear term, such as $f(u)$, weak convergence is not enough: if $u_m \rightharpoonup u$ weakly, $f(u_m)$ need not converge to $f(u)$ (4A.7 Weak Convergence and the Direct Method, the oscillation examples). One needs strong convergence in some norm, and in space-time the source of strong convergence is a compactness lemma that combines a bound on spatial regularity with a bound on the time derivative. It starts from an inequality between three spaces.

We use it with $X_0 = H^1_0$, $X = L^2$, $X_1 = H^{-1}$. The inclusion $H^1_0 \hookrightarrow L^2$ is compact by Rellich (4A.11 Sobolev Embeddings and Critical Exponents), and the inclusion $L^2 \hookrightarrow H^{-1}$ is also compact: it is the adjoint of the first, and adjoints of compact operators on Hilbert spaces are compact (4A.8 Compact Operators and Spectra).

The lemma's message: spatial compactness (a bound in a compactly embedded space) plus control of time oscillation (a bound on the time derivative, even in a weak norm) gives compactness in space-time. Neither alone suffices: functions bounded in $L^2(H^1_0)$ can oscillate wildly in time, like $\sin(mt)\phi(x)$, and functions with bounded time derivatives can oscillate in space.

11.6History

Jacques-Louis Lions developed the Hilbert-space theory of evolution equations, with Gelfand triples and Bochner spaces, in the 1950s and 1960s; his book with Enrico Magenes (1968–70) and his 1969 book on nonlinear problems systematised it, including the compactness lemma he proved with Jean-Pierre Aubin (Aubin's paper is from 1963). Gunnar Ehrling's lemma dates from 1954. Jean Leray constructed weak solutions of the Navier–Stokes equations in 1934 and Eberhard Hopf extended the construction to bounded domains in 1951. Takeru Higuchi's paper on drug release appeared in 1961; John Crank's The Mathematics of Diffusion (1956, second edition 1975) collected the solutions used in the release calculation.

11.7Exercises

Solution

(a) The sine coefficients of $1$ on $(0, L)$ are $\frac2L\int_0^L\sin\frac{k\pi x}{L}dx = \frac{2}{k\pi}(1 - (-1)^k)$, which is $\frac{4}{k\pi}$ for odd $k$. (b) $\int_0^L\sin\frac{k\pi x}{L}dx = \frac{2L}{k\pi}$ for odd $k$, so the remaining mass is $\frac{8L}{\pi^2}\sum_{k\text{ odd}}\frac{1}{k^2}e^{-k^2\pi^2Dt/L^2}$, equal to $L$ at $t = 0$. (c) Near a face, at early times, the concentration is approximately $u = \operatorname{erf}\big(\frac{x}{2\sqrt{Dt}}\big)$ ($0$ at the face, $1$ inside), so $u_x(0, t) = \frac{1}{\sqrt{\pi Dt}}$, so the flux out is $Du_x = \frac{D}{\sqrt{\pi Dt}} = \sqrt{\frac{D}{\pi t}}$, and $\int_0^t\sqrt{\frac{D}{\pi s}}ds = 2\sqrt{\frac{Dt}{\pi}}$ per face.

Solution

$\|u(t)\|^2 = \frac\pi2\sum b_k^2e^{-2k^2t}$ and $\|\nabla u\|^2 = \frac\pi2\sum k^2b_k^2e^{-2k^2t}$, so $2\int_0^t\|\nabla u\|^2 = \frac\pi2\sum b_k^2(1 - e^{-2k^2t})$. Adding gives $\frac\pi2\sum b_k^2 = \|g\|^2$.

Solution

$u = e^{-N^2t}\sin Nx$, $\|\nabla u\| = Ne^{-N^2t}\sqrt{\pi/2}$, $\|g\| = \sqrt{\pi/2}$. With $s = N\sqrt t$, $\sqrt t\,Ne^{-N^2t} = se^{-s^2}$, maximised at $s = \frac{1}{\sqrt2}$ with value $\frac{1}{\sqrt{2e}}$. So for each $t$ some data (with $N \approx (2t)^{-1/2}$) nearly attain $\frac{1}{\sqrt{2et}}\|g\|$.

Solution

Continuity at the corner gives $g(0) = u(0, 0) = 0$. If $u_t$ is continuous up to the corner, then $u_t(0, t) = 0$ for $t > 0$ (as $u(0, t) = 0$), so $u_t(0, 0) = 0$; but $u_t = u_{xx}$, so $g''(0) = 0$. For $g \equiv 1$, near the corner $u \approx \operatorname{erf}\big(\frac{x}{2\sqrt t}\big)$, whose $t$-derivative is $-\frac{x}{2\sqrt\pi t^{3/2}}e^{-x^2/4t}$, unbounded near $(0, 0)$ (along $x = \sqrt t$ it is of size $t^{-1}$).

Solution

(a) $\|u_m\|_{L^2(H^1_0)}^2 = \|\phi\|_{H^1_0}^2\int_0^T\sin^2(mt)dt \leq T\|\phi\|^2_{H^1_0}$. But $\sin(mt) \rightharpoonup 0$ weakly in $L^2(0, T)$ while $\int\sin^2(mt) \to \frac T2$, so no subsequence converges strongly (the only possible limit is $0$). The time derivative $m\cos(mt)\phi$ is unbounded. (b) $\|\sin mx\|_{H^1_0} = m\sqrt{\pi/2}$ is unbounded; the spatial bound fails, and $\sin mx \rightharpoonup 0$ without strong convergence.

Solution

(a) $\frac{d}{dt}\int|T|^2 = 2\int\langle\Delta T + Q(T), T\rangle$ and $\int\langle\Delta T, T\rangle = -\int|\nabla T|^2$ on a closed manifold; $|\langle Q(T), T\rangle| \leq C|T|^3 \leq C\sup|T|\,|T|^2$. (b) Dropping the gradient term, $\frac{d}{dt}\int|T|^2 \leq 2CK\int|T|^2$, so Gronwall gives the exponential bound. Integrating the full inequality over $[0, t]$, $2\int_0^t\int|\nabla T|^2 \leq \int|T(0)|^2 + 2CK\int_0^t\int|T|^2$, which is bounded.