Power Series and the Cauchy–Kovalevskaya Theorem

The simplest idea for solving a differential equation is to write the solution as a power series and determine its coefficients one at a time. For ODEs this works whenever the coefficients are analytic, and it is how Bessel's and Legendre's functions were first found (6B.4 Green’s Functions, Separation of Variables and Transforms). For PDEs it works too, under one condition on the surface carrying the data: the Cauchy–Kovalevskaya theorem says that a PDE with analytic coefficients and analytic data on a noncharacteristic surface has a unique analytic solution near that surface. It is the oldest general existence theorem for PDE, and the only one that applies to equations of every type at once.

It is also the clearest example of a theorem that answers the wrong question. It says nothing about stability: Hadamard's ill-posed problem for the Laplace equation (6A.1 What a PDE Is) satisfies all its hypotheses. It says nothing about non-analytic data: Hans Lewy found in 1957 a simple linear equation with smooth coefficients that has no solutions at all for most smooth right-hand sides. And for the heat equation with initial data it does not apply, because the initial surface is characteristic, and Sofia Kovalevskaya herself showed that the formal power series can diverge. These failures are why modern PDE theory works in Sobolev spaces with energy estimates. This chapter proves the theorem carefully and then explains, with examples, exactly what it does not do.

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

• work with real analytic functions of several variables, multi-index notation and majorant series;
• decide whether a surface is characteristic for a linear equation, and show how noncharacteristic data determine all derivatives on the surface;
• prove the Cauchy–Kovalevskaya theorem for first-order systems by the method of majorants;
• explain Kovalevskaya's heat-equation example, Hadamard's instability, Holmgren's uniqueness theorem and Lewy's unsolvable equation;
• prove that solutions of the heat equation are analytic in space at positive times.

5.1Continuing a potential field downward

5.2Real analytic functions

A function $f : U \to \mathbb{R}$ on an open set $U \subseteq \mathbb{R}^n$ is real analytic near $x_0 \in U$ if there are $r > 0$ and constants $f_\alpha$ such that

the series converging absolutely. Here $\alpha = (\alpha_1, \ldots, \alpha_n)$ runs over multi-indices of non-negative integers, $x^\alpha = x_1^{\alpha_1}\cdots x_n^{\alpha_n}$, $|\alpha| = \alpha_1 + \cdots + \alpha_n$, $\alpha! = \alpha_1!\cdots\alpha_n!$ and $\partial^\alpha = \partial_1^{\alpha_1}\cdots\partial_n^{\alpha_n}$. A convergent power series can be differentiated term by term inside its region of convergence, so $f$ is smooth and $f_\alpha = \frac{\partial^\alpha f(x_0)}{\alpha!}$: the series is the Taylor series. Polynomials, exponentials, $\sin$, $\cos$, and $\frac{1}{1 - x}$ near $0$ are analytic; so are sums, products, quotients (where the denominator is non-zero) and compositions of analytic functions, and harmonic functions (6A.2 Harmonic Functions). The function $e^{-1/x^2}$ (with value $0$ at $0$) is smooth but not analytic at $0$: all its derivatives vanish there, so its Taylor series is zero, yet the function is not.

A useful test: $f$ is analytic near $x_0$ if and only if its derivatives satisfy Cauchy-type estimates $|\partial^\alpha f(x)| \leq C\frac{|\alpha|!}{\rho^{|\alpha|}}$ uniformly near $x_0$, for some $C$ and $\rho > 0$. The "if" direction is the remainder estimate of 6A.2 Harmonic Functions's proof that harmonic functions are analytic.

Majorants. For two power series $f = \sum f_\alpha x^\alpha$ and $g = \sum g_\alpha x^\alpha$, say $g$ majorises $f$, written $g \gg f$, if $g_\alpha \geq |f_\alpha|$ for every $\alpha$. If $g$ converges for $|x| < r$ and $g \gg f$, then $f$ converges there too (each term of $f$ is dominated in absolute value by the corresponding term of $g$ evaluated at $(|x_1|, \ldots, |x_n|)$). The basic majorant is a geometric series.

5.3Noncharacteristic surfaces

Consider a linear equation of order $k$,

with Cauchy data on a smooth hypersurface $\Gamma$ with unit normal $\nu$: the values of $u$ and of its normal derivatives up to order $k - 1$, that is, $u, \partial_\nu u, \ldots, \partial_\nu^{k-1}u$ on $\Gamma$. Can these data and the equation determine all the derivatives of $u$ on $\Gamma$, so that the Taylor series of $u$ at each point of $\Gamma$ is known?

Flatten $\Gamma$ to the plane $\{x_n = 0\}$ by a change of coordinates, so $\nu = e_n$. The data then give every derivative $\partial^\beta u$ on $\Gamma$ with $\beta_n \leq k - 1$ (differentiate the data tangentially). The only $k$-th order derivative not yet known is $\partial_n^ku$, and the equation contains it with the coefficient $a_{(0,\ldots,0,k)}$. If that coefficient is non-zero, the equation can be solved for it:

Differentiating the equation and repeating determines all higher normal derivatives too (Exercise 5.5). In general coordinates the coefficient is the principal symbol evaluated on the normal.

The expression is the principal symbol $\sigma(x_0, \nu)$ of 6A.1 What a PDE Is (up to factors of $i$). So:

• For an elliptic operator, $\sigma(x, \nu) \neq 0$ for every $\nu \neq 0$: no surface is characteristic. This is why Cauchy–Kovalevskaya applies to the Laplace equation with data on any surface, and also why the theorem can't tell good problems from bad ones.
• For the wave equation $u_{tt} - \Delta u$, with $\nu = (\nu_x, \nu_t)$, the symbol is $\nu_t^2 - |\nu_x|^2$; the characteristic surfaces are those whose normal lies on the light cone, such as the cones $|x| = t$. Spacelike surfaces, such as $t = 0$, are noncharacteristic.
• For the heat equation $u_t - \Delta u$, the principal part is $-\Delta$ (order $2$ in $x$ only), and the symbol is $-|\nu_x|^2$. The initial surface $t = 0$ has normal $\nu = (0, 1)$ and symbol $0$: it is characteristic. Data at $t = 0$ give $u$, and the equation then gives $u_t = \Delta u$, but not by solving for a second $t$-derivative; the time derivatives are determined by differentiating the data in $x$, which loses derivatives, and the resulting formal series can diverge (Exercise 5.8).

5.4The Cauchy–Kovalevskaya theorem

By introducing all lower-order derivatives as new unknowns, any noncharacteristic problem for a $k$-th order system can be reduced to a first-order system of the form below (Evans, section 4.6.3, carries out the reduction). So it suffices to prove:

The form is less special than it looks. The data were made zero by subtracting any analytic function with the given data; the coefficients do not depend on $x_n$, which can be arranged by adding $x_n$ as an extra unknown $u^{m+1}$ with $\partial_nu^{m+1} = 1$ and zero data; and the equation is solved for $\partial_n\mathbf u$, which is what noncharacteristic means.

The proof is a comparison argument, like the maximum principle, but at the level of Taylor coefficients: an explicitly solvable problem with larger coefficients has a larger (convergent) solution, and so the original converges. The non-negativity of the recursion's coefficients plays the role that positivity of the heat kernel played in 6A.5 Maximum Principles.

5.5What the theorem does not say

It is local. The solution exists in a neighbourhood whose size depends on the radii of convergence of the data and coefficients, and it may not extend further, as for the ODE $y' = y^2$ (2B.10 Ordinary Differential Equations).

It gives no stability. Hadamard's problem, $u_{xx} + u_{yy} = 0$ with $u(x, 0) = 0$ and $u_y(x, 0) = \frac1n\sin nx$, has analytic data on a noncharacteristic line, so Cauchy–Kovalevskaya applies; the solution $\frac{1}{n^2}\sin nx\sinh ny$ exists, is analytic and is unique, and is huge at $y = 1$ although the data are tiny (6A.1 What a PDE Is). Existence and uniqueness without continuous dependence is useless for computation, as the downward-continuation box explained.

Uniqueness extends beyond analytic solutions. Step 1 only shows uniqueness among analytic solutions. Erik Holmgren proved in 1901 that for linear equations with analytic coefficients, a $C^k$ solution with zero Cauchy data on a noncharacteristic surface vanishes near it: there are no non-analytic solutions with the same data either. The proof uses Cauchy–Kovalevskaya for the adjoint equation to build enough test functions (Taylor I, §6.4; John, Partial Differential Equations). For the wave equation, Holmgren's theorem gives uniqueness in the domain of dependence, which 6B.3 The Wave Equation proved with energy.

It says nothing about non-analytic data, and smooth data can have no solution. Hans Lewy found in 1957 that the linear operator

on $\mathbb{R}^3$, with simple polynomial coefficients, has the property that for most smooth (indeed $C^\infty$) functions $f$, the equation $Lu = f$ has no solution, even a distributional one, in any open set. It is not a pathology of bad coefficients: $L$ is a natural operator in complex analysis (the tangential Cauchy–Riemann operator on the boundary of a domain in $\mathbb{C}^2$). By contrast, Bernard Malgrange and Leon Ehrenpreis proved in 1954–55 that every linear operator with constant coefficients has a fundamental solution, and so is locally solvable (Taylor I, §3.10). Lewy's example showed that local solvability for variable coefficients is a genuine question, answered for large classes of operators by Lars Hörmander, Louis Nirenberg and François Trèves in the 1960s and 70s.

It does not apply to the heat equation's initial-value problem, whose initial surface is characteristic, and there the formal series can diverge (Exercise 5.8). Yet the heat equation's solutions are as smooth as can be: for $t > 0$ they are analytic in $x$, for any bounded initial data (Exercise 5.10). Analyticity is produced by the equation, not assumed.

The lesson the subject drew from these examples is that the analytic category is the wrong setting: too rigid to contain the solutions of interest, and blind to stability. Existence should be proved in spaces defined by norms, with estimates that give continuous dependence: energy spaces for hyperbolic equations (6B.14 Hyperbolic Equations and Systems), Sobolev and Hölder spaces for elliptic and parabolic ones (6A.6 Weak Solutions and Elliptic Regularity, 6B.7 Schauder Theory and the Continuity Method).

5.6History

Augustin-Louis Cauchy proved existence for analytic ODE and for some first-order PDE by majorants in the 1840s ("calcul des limites"). Sofia Kovalevskaya proved the general theorem in her 1874 doctoral thesis at Göttingen, supervised by Karl Weierstrass, and gave the example showing that the heat equation's formal series can diverge; she was among the first women to earn a doctorate in mathematics, and became the first woman to hold a full professorship in mathematics in northern Europe (Stockholm, 1889). Erik Holmgren's uniqueness theorem dates from 1901. Jacques Hadamard's analysis of the ill-posed Cauchy problem appeared in 1902 and in his 1923 lectures. Hans Lewy's example was published in 1957, and the Malgrange–Ehrenpreis theorem in 1954–55.

5.7Exercises

Solution

(a) The coefficients of $\sin$, $\cos$ and $e^x$ are at most $1$ in absolute value, and those of $\frac{1}{1 - x}$ are all $1$. $\frac{1}{2 - x} = \sum\frac{x^k}{2^{k+1}}$, which equals $\frac{Cs}{s - x}$ with $s = 2$, $C = \frac12$. (b) The coefficients of $fh$ are $\sum_{\beta+\gamma=\alpha}f_\beta h_\gamma$, bounded in absolute value by $\sum g_\beta k_\gamma$, the coefficients of $gk$; sums are immediate.

Solution

$u_{yy} = -u_{xx}$, so $\partial_y^{2k}u = (-\partial_x^2)^ku$ and $\partial_y^{2k+1}u = (-\partial_x^2)^k\partial_yu$; evaluate at $y = 0$. The series is $u = \sum_k(-1)^k\big(f^{(2k)}(x)\frac{y^{2k}}{(2k)!} + g^{(2k)}(x)\frac{y^{2k+1}}{(2k+1)!}\big)$. With $g = \frac1n\sin nx$, $g^{(2k)} = (-1)^kn^{2k-1}\sin nx$, giving $\frac1n\sin nx\sum\frac{n^{2k}y^{2k+1}}{(2k + 1)!} = \frac{1}{n^2}\sin nx\sinh ny$.

Hint

Write $Q = \sqrt{(r - s)^2 - 2mnCrx_n}$. Then $r - s - mv = \frac{(n - 1)(r - s) + Q}{n}$, $\partial_nv = \frac{Cr}{Q}$, and each $\partial_jv = \frac{1}{mn}\big(-1 + \frac{r - s}{Q}\big)$ for $j < n$.

Solution

Using the hint, $m\sum_{j<n}\partial_jv + 1 = \frac{n - 1}{n}\big(\frac{r - s}{Q} - 1\big) + 1 = \frac{(n - 1)(r - s) + Q}{nQ}$. So the right side is $\frac{Cr\,n}{(n - 1)(r - s) + Q}\cdot\frac{(n - 1)(r - s) + Q}{nQ} = \frac{Cr}{Q} = \partial_nv$. At $x_n = 0$, $Q = r - s$ and $v = 0$.

Solution

(a) Normal $\frac{(1, 1)}{\sqrt2}$, symbol $\nu_x^2 - \nu_y^2 = 0$: characteristic. (b) Normal $(0, 1)$: symbol $-1 \neq 0$: noncharacteristic. (c) Normal $(0, 1)$ in $(x, t)$: $\nu_t^2 - |\nu_x|^2 = 1$: noncharacteristic. (d) Normal proportional to $(-x/|x|, 1)$: $1 - 1 = 0$: characteristic. (e) Normal $e_1$: the symbol of the principal part $-\Delta$ is $-|\nu_x|^2 = -1 \neq 0$: noncharacteristic (data on a timelike plane are fine; it is the initial plane $t = 0$ that is characteristic).

Solution

(a) $\partial_t^ku = \partial_x^{2k}u$, evaluated at $t = 0$. (b) The coefficient of $x^{2k}$ is $(-1)^k$, so $g^{(2k)}(0) = (-1)^k(2k)!$. (c) The coefficients $\frac{(2k)!}{k!}$ grow faster than any geometric sequence ($\frac{(2k + 2)!/(k + 1)!}{(2k)!/k!} = 2(2k + 1) \to \infty$), so by the ratio test the series diverges for all $t \neq 0$.

Solution

The equation is linear with constant (hence analytic) coefficients, and $t = 0$ is noncharacteristic, so Holmgren gives vanishing in a neighbourhood of the segment. Applied repeatedly on a family of noncharacteristic surfaces sweeping out the triangle (curves that bend up towards its apex while staying spacelike), it gives the whole domain of dependence, the same conclusion as the energy argument. The energy argument is more robust: it works for non-analytic coefficients.

Solution

(a) $\widehat{\partial_x^ku} = (2\pi i\xi)^ke^{-4\pi^2\xi^2t}\hat g$, and $\sup_y y^ke^{-y^2t}$ is attained at $y^2 = \frac{k}{2t}$, with value $\big(\frac{k}{2t}\big)^{k/2}e^{-k/2}$. (b) From $k^k \leq e^kk!$, $k^{k/2} \leq e^{k/2}\sqrt{k!}$, so $\big(\frac{k}{2et}\big)^{k/2} \leq (2t)^{-k/2}\sqrt{k!}$. The Sobolev bound holds because $w(x)^2 = \int_{-\infty}^x2ww' \leq 2\|w\|_2\|w'\|_2 \leq \|w\|_2^2 + \|w'\|_2^2$. Then $\sup|\partial_x^ku| \leq \|\partial_x^ku\|_2 + \|\partial_x^{k+1}u\|_2 \leq \big((2t)^{-k/2}\sqrt{k!} + (2t)^{-(k+1)/2}\sqrt{(k + 1)!}\big)\|g\|_2$, and for $2t \leq 1$ each term is at most $(2t)^{-(k+1)/2}\sqrt{(k + 1)!}\|g\|_2$. (For larger $t$, start the flow at time $t - \frac12$ instead.) (c) The Taylor coefficients satisfy $\frac{|\partial^ku(x_0)|}{k!} \leq C\frac{\sqrt{(k + 1)!}}{k!}(2t)^{-(k+1)/2}$, and $\frac{\sqrt{(k+1)!}}{k!}R^k \to 0$ for every $R$, so the series converges everywhere; the remainder in Taylor's theorem is bounded by the next term's size on any interval, so the series converges to $u$.