Semigroups and Evolution Equations

6B.12 Unique Continuation and Backward Uniqueness proved backward uniqueness for $u' = Au$ by treating the heat equation as an ordinary differential equation in a Hilbert space, with the Laplacian as a self-adjoint operator, and used the fact that its solutions are smooth in time. This chapter builds that point of view in general. An ODE $y' = ay$ is solved by $y(t) = e^{ta}y(0)$, and a linear system $y' = Ay$ by the matrix exponential $e^{tA}y(0)$. Many evolution PDE can be written the same way: $u' = Au$, where $u(t)$ lives in a function space and $A$ is a differential operator, such as the Laplacian. The heat equation becomes $u' = \Delta u$, the wave equation the first-order system $(u, v)' = (v, \Delta u)$, a transport equation $u' = b\cdot\nabla u$. The solution should be "$e^{tA}u(0)$". The difficulty is that $A$ is unbounded, so the power series $\sum\frac{t^kA^k}{k!}$ makes no sense. Semigroup theory says when $e^{tA}$ nevertheless exists, as a family of bounded operators $S(t)$ with $S(t + s) = S(t)S(s)$, and identifies exactly which operators $A$ generate such families: the Hille–Yosida theorem.

The payoff is a single framework for existence, uniqueness and continuous dependence of linear evolution equations, and a clean form of Duhamel's principle, $u(t) = S(t)g + \int_0^tS(t - s)f(s)\,ds$, which is the starting point of every fixed-point argument for nonlinear equations (6A.8 Nonlinear Parabolic Equations). It also connects PDE with probability: a Markov process is described by a semigroup of operators on functions, its generator is a differential operator, and the semigroup law is the Chapman–Kolmogorov equation (6D.3 Stochastic Differential Equations and Feynman–Kac).

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

• define strongly continuous and contraction semigroups and their generators, and prove the basic properties of generators;
• prove that the resolvent is the Laplace transform of the semigroup, and prove the Hille–Yosida theorem by Yosida approximation;
• apply it to the heat equation, the wave equation and the transport equation, using Lax–Milgram for the resolvent;
• prove the smoothing estimate $\|AS(t)\| \leq \frac{C}{t}$ for self-adjoint generators, and use it to handle nonlinearities with derivatives;
• write mild solutions by Duhamel's formula, and prove the Lie–Trotter product formula behind operator splitting.

13.1Credit ratings as a Markov semigroup

13.2Semigroups and their generators

Let $X$ be a Banach space and $L(X)$ the bounded linear operators on it (4A.1 Banach Spaces and Bounded Operators).

For a matrix $A$, $S(t) = e^{tA}$ is a strongly continuous semigroup with generator $A$ and $D(A) = X$. In infinite dimensions $D(A)$ is usually a proper dense subspace: for the heat semigroup on $L^2$, the limit $\frac{S(t)u - u}{t}$ exists in $L^2$ exactly when $\Delta u \in L^2$ in a suitable sense.

So for $u \in D(A)$, the function $u(t) = S(t)u$ solves $u' = Au$, $u(0) = u$: the abstract Cauchy problem. It is the only solution: if $v$ is another, then $\frac{d}{ds}S(t - s)v(s) = -AS(t - s)v(s) + S(t - s)Av(s) = 0$, so $v(t) = S(t)v(0)$.

The resolvent is a Laplace transform. The scalar identity $\int_0^\infty e^{-\lambda t}e^{ta}\,dt = \frac{1}{\lambda - a}$ (for $\lambda > a$) has an operator version, which is the bridge between a semigroup and its generator.

13.3The Hille–Yosida theorem

The proposition lists necessary conditions on a generator. Remarkably, they are also sufficient.

The Yosida approximation $A_\lambda = \lambda^2R_\lambda - \lambda$ is a smoothed version of $A$: for the Laplacian, $A_\lambda = \Delta(I - \lambda^{-1}\Delta)^{-1}$ multiplies the frequency-$\xi$ mode by $\frac{-4\pi^2|\xi|^2}{1 + 4\pi^2|\xi|^2/\lambda}$, bounded by $\lambda$, close to $-4\pi^2|\xi|^2$ for $|\xi|^2 \ll \lambda$. In numerical analysis, $(I - \frac{t}{n}A)^{-n} = (\frac nt R_{n/t})^n$ is the implicit Euler scheme, and it converges to $S(t)$ too (Pazy, Theorem 1.8.3): the semigroup is the limit of implicit time-stepping, which is how parabolic equations are computed (6D.9 Finite Difference Methods).

Dissipative operators. In a Hilbert space the resolvent condition has a convenient form. An operator $A$ is dissipative if $\operatorname{Re}\langle Au, u\rangle \leq 0$ for all $u \in D(A)$; then $\|(\lambda - A)u\|\|u\| \geq \operatorname{Re}\langle(\lambda - A)u, u\rangle \geq \lambda\|u\|^2$, so $\|(\lambda - A)u\| \geq \lambda\|u\|$, which is $\|R_\lambda\| \leq \frac1\lambda$ wherever $R_\lambda$ is defined. The Lumer–Phillips theorem (1961) says: a densely defined dissipative operator with $\lambda - A$ onto for some $\lambda > 0$ generates a contraction semigroup. So, in practice: check dissipativity by an integration by parts, and surjectivity by solving an elliptic problem.

13.4Applications

The heat equation. Let $U$ be bounded with smooth boundary, $X = L^2(U)$, and $A = \Delta$ with $D(A) = H^2(U)\cap H^1_0(U)$. Dissipative: $\langle\Delta u, u\rangle = -\int|\nabla u|^2 \leq 0$. Onto: for $\lambda > 0$ and $f \in L^2$, the problem $\lambda u - \Delta u = f$, $u \in H^1_0$, has a unique weak solution by Lax–Milgram (the form $\int\lambda uv + \nabla u\cdot\nabla v$ is coercive), and it lies in $H^2$ by elliptic regularity (6A.6 Weak Solutions and Elliptic Regularity). Closed and densely defined: $D(A)$ contains $C_c^\infty$, and closedness follows from the elliptic estimate $\|u\|_{H^2} \leq C(\|\Delta u\|_{L^2} + \|u\|_{L^2})$. So $\Delta$ generates a contraction semigroup $S(t) = e^{t\Delta}$ on $L^2(U)$, and $u(t) = S(t)g$ is the weak solution of 6A.7 Parabolic Regularity and 6B.11 Linear Parabolic Equations in Full. In eigenfunctions, $S(t)g = \sum e^{-\lambda_kt}(g, \phi_k)\phi_k$. The same works for any uniformly elliptic $-L$ with $c \geq 0$, and on $L^p$ for $1 < p < \infty$ using the $L^p$ theory of 6B.8 Lᵖ Theory and Strong Solutions.

The wave equation. Write $u_{tt} = \Delta u$ as the system $\frac{d}{dt}(u, v) = (v, \Delta u)$ on $X = H^1_0(U)\times L^2(U)$, with the energy inner product $\langle(u, v), (\tilde u, \tilde v)\rangle = \int\nabla u\cdot\nabla\tilde u + v\tilde v$, and $A(u, v) = (v, \Delta u)$, $D(A) = (H^2\cap H^1_0)\times H^1_0$. Then

$A$ is skew, so both $A$ and $-A$ are dissipative, and the resolvent equations reduce to $\lambda^2u - \Delta u = \lambda f + g$, solvable by Lax–Milgram. So $A$ generates a group of isometries $S(t)$, $t \in \mathbb{R}$: conservation of energy and time reversibility (6B.3 The Wave Equation), now as abstract theorems.

Transport. On $X = L^2(\mathbb{R})$, the translations $S(t)u(x) = u(x + t)$ form a contraction semigroup (strongly continuous by the continuity of translation in $L^2$, 3A.8 Convolution and Mollifiers), with generator $Au = u'$ and $D(A) = H^1(\mathbb{R})$ (Exercise 13.8). The solution of $u_t = u_x$ is $u(x + t)$, the transport equation of 6B.1 First-Order Equations and Characteristics.

13.5Analytic semigroups and smoothing

The heat semigroup does more than solve the equation: it smooths. In semigroup language, $S(t)$ maps all of $X$ into $D(A)$ for $t > 0$, with a bound on $AS(t)$ that blows up at a definite rate as $t \to 0$. Semigroups with this property, which extend holomorphically to a sector of complex times, are called analytic. For self-adjoint generators the estimate is a one-line spectral computation.

For the Dirichlet Laplacian, $\|(-\Delta)^{1/2}u\| = \|\nabla u\|$, so the second bound is $\|\nabla S(t)g\| \leq \frac{\|g\|}{\sqrt{2et}}$: the gradient smoothing of 6A.3 The Heat Equation on ℝⁿ and 6B.11 Linear Parabolic Equations in Full, with the parabolic rate, now as an operator norm. The wave group has no such estimate ($\|AS(t)\|$ is infinite for every $t$), which is the abstract form of "waves do not smooth" (6B.3 The Wave Equation).

13.6Duhamel's formula and mild solutions

For the inhomogeneous problem $u' = Au + f(t)$, $u(0) = g$, variation of constants suggests

For $g \in X$ and $f \in L^1(0, T; X)$ this formula always defines a continuous function, called the mild solution. If $g \in D(A)$ and $f \in C^1([0, T]; X)$, it is a genuine (classical) solution, differentiable with $u' = Au + f$ (Pazy, Theorem 4.2.4; Evans, section 7.4.3). For semilinear equations $u' = Au + F(u)$ one looks for a fixed point of the map $u \mapsto S(t)g + \int_0^tS(t - s)F(u(s))\,ds$, which is how 6A.8 Nonlinear Parabolic Equations solved $u_t = \Delta u + f(u)$; with the smoothing estimate, nonlinearities that involve derivatives can be handled too, because $\int_0^t\|(-A)^{1/2}S(t - s)\|\,ds \leq \int_0^t\frac{ds}{\sqrt{2e(t - s)}}$ is finite (Exercise 13.12). That integrability, $(t - s)^{-1/2}$ but not $(t - s)^{-1}$, is the reason a parabolic equation can absorb a nonlinearity with one derivative but not two, and it is the structural reason quasilinear equations need a different method (6C.12 Quasilinear Parabolic Systems).

13.7Splitting: the Lie–Trotter formula

Many evolution equations are sums of simpler ones: diffusion plus reaction, diffusion plus drift. If $e^{tA}$ and $e^{tB}$ are easy to compute separately, alternating them in small steps approximates $e^{t(A+B)}$.

The error comes from non-commutation: the $O(n^{-2})$ term in $X_n - Y_n$ is $\frac{t^2}{2n^2}[B, A]$, and if $A$ and $B$ commute the formula is exact for every $n$. The symmetric Strang splitting $e^{tA/2n}e^{tB/n}e^{tA/2n}$ cancels that term and has error $O(n^{-2})$. Trotter (1959) and Kato proved versions for unbounded generators, and Taylor III, §15.5, proves a nonlinear one for reaction–diffusion systems; it is also how the Feynman–Kac formula can be derived from a path integral (6D.3 Stochastic Differential Equations and Feynman–Kac). In computational finance and physics, splitting is the standard way to solve equations like $u_t = \frac12\sigma^2u_{xx} + \mu u_x - ru$: each part is solved exactly or by a simple scheme, and the parts are alternated (6D.9 Finite Difference Methods).

13.8History

Einar Hille (1948) and Kōsaku Yosida (1948) proved the generation theorem independently. William Feller (1952) and Ralph Phillips extended it, and Günter Lumer and Phillips proved their theorem in 1961. Hale Trotter's product formula appeared in 1959, generalising Sophus Lie's formula for matrices. Semigroup methods for nonlinear parabolic equations were developed by Tosio Kato, Hiroshi Fujita and others from the 1960s; Amnon Pazy's book (1983) became the standard reference. John Kingman studied the embedding problem for Markov matrices in 1962, and Jarrow, Lando and Turnbull's credit model appeared in 1997.

13.9Exercises

Solution

(a) $Q\mathbf 1 = 0$, so $e^{tQ}\mathbf 1 = \sum\frac{t^kQ^k}{k!}\mathbf 1 = \mathbf 1$. (b) For $c \geq \max_i|q_{ii}|$, $Q + cI$ has non-negative entries, so does every power, so $e^{t(Q+cI)} \geq 0$ entrywise; and $e^{tQ} = e^{-ct}e^{t(Q+cI)}$ since $cI$ commutes with $Q$. (c) The eigenvalues of $Q$ are $0$ and $-(a + b)$; $P(t) = \frac{1}{a + b}\begin{pmatrix}b + ae^{-(a+b)t} & a - ae^{-(a+b)t}\\ b - be^{-(a+b)t} & a + be^{-(a+b)t}\end{pmatrix}$, which tends to the matrix with both rows $\big(\frac{b}{a+b}, \frac{a}{a+b}\big)$, the stationary distribution.

Solution

(b) $\big|\frac{e^{2\pi it\xi} - 1}{t} - 2\pi i\xi\big| \to 0$ pointwise and is bounded by $4\pi|\xi|$, so dominated convergence with $\int|\xi|^2|\hat u|^2 < \infty$ gives $L^2$ convergence to $\widehat{u'}$. (c) $R_\lambda f(x) = \int_0^\infty e^{-\lambda t}f(x + t)\,dt$, convolution of $f$ with the function $e^{\lambda y}\mathbf 1_{y<0}$ (reflected), whose $L^1$ norm is $\frac1\lambda$; by Young's inequality (3A.8 Convolution and Mollifiers) $\|R_\lambda f\|_2 \leq \frac1\lambda\|f\|_2$. Check: $\lambda R_\lambda f - (R_\lambda f)' = f$ by differentiating under the integral and integrating by parts.

Solution

(a) $\int_0^\pi u''u = -\int_0^\pi u'^2$. (b) If $f = \sum f_k\sin kx$, then $u = \sum\frac{f_k}{\lambda + k^2}\sin kx$, so $\|R_\lambda\| = \sup_k\frac{1}{\lambda + k^2} = \frac{1}{\lambda + 1}$. The improvement reflects the spectral gap: the largest eigenvalue of $A$ is $-1$, not $0$, so $e^{tA}$ decays like $e^{-t}$ and $R_\lambda$ is better than for a general contraction semigroup.

Solution

$R_\lambda = \frac{1}{\lambda - a}$, $A_\lambda = \frac{\lambda a}{\lambda - a} \to a$, so $e^{tA_\lambda} \to e^{ta}$. On $\sin kx$, $A = -k^2$, so $A_\lambda\sin kx = \frac{-\lambda k^2}{\lambda + k^2}\sin kx$: close to $-k^2$ for $k^2 \ll \lambda$, bounded by $-\lambda$ for large $k$. The Yosida approximation caps the decay rate of high frequencies at $\lambda$.

Solution

To second order, $e^{hA}e^{hB} = I + h(A + B) + \frac{h^2}{2}(A^2 + 2AB + B^2) + O(h^3)$ and $e^{h(A+B)} = I + h(A + B) + \frac{h^2}{2}(A^2 + AB + BA + B^2) + O(h^3)$; the difference is $\frac{h^2}{2}(AB - BA)$. For the symmetric product $(I + \frac{hA}{2} + \frac{h^2A^2}{8})(I + hB + \frac{h^2B^2}{2})(I + \frac{hA}{2} + \frac{h^2A^2}{8})$, the $h^2$ terms are $\frac{A^2}{8} + \frac{B^2}{2} + \frac{A^2}{8}$ from single factors and $\frac{AB}{2} + \frac{A^2}{4} + \frac{BA}{2}$ from pairs, totalling $\frac12(A^2 + AB + BA + B^2)$, which matches. With local error $O(h^3)$ per step and $n$ steps of size $h = t/n$, the telescoping argument of Theorem 13.6 gives total error $O(n\cdot n^{-3}) = O(n^{-2})$.

Solution

(a) On mean-zero functions the eigenvalues of $-A$ are $k^2 \geq 1$ with eigenfunctions $e^{ikx}$, and $|\partial_x|$ multiplies $e^{ikx}$ by $|k| = (k^2)^{1/2}$, so $\|\partial_xS(t)\| = \|(-A)^{1/2}S(t)\| \leq (2et)^{-1/2}$. (The mean is conserved by the equation and can be handled separately.) (b) For $u$, $v$ in a ball of radius $R$, $\|u^2 - v^2\| \leq 2R\|u - v\|$, so the difference of the integrals is at most $\frac12\int_0^t\frac{2R\|u - v\|}{\sqrt{2e(t - s)}}ds \leq \frac{2R\sqrt t}{\sqrt{2e}}\sup\|u - v\|$, less than half for small $T$; similarly the map sends the ball to itself. (c) Two derivatives would require $\|(-A)S(t - s)\| \sim \frac{1}{t - s}$, which is not integrable at $s = t$: the mild formulation loses control. In a quasilinear equation the highest derivatives appear in the nonlinearity, so one instead treats the leading part as a variable-coefficient linear operator, freezes it at the solution, and uses Schauder estimates for that linear operator.