the Cauchy-Kovalevskaya theorem
/ Kovalevskaya -> koh-vah-LYEV-ska-ya; Cauchy -> koh-SHEE /
Here is the most general 'a solution exists' statement in classical PDE theory, and it is reassuringly simple in spirit. If everything in sight is analytic — the equation, its coefficients, and the data you prescribe can all be written as convergent power series — then the Cauchy problem has a solution, at least in a small neighbourhood of the surface where you laid the data. It is the local existence theorem that underlies the very idea that posing Cauchy data is a sensible thing to do. Named for Augustin-Louis Cauchy and Sofya Kovalevskaya, who gave the first rigorous and general proof.
The mechanism is power series, and it is almost mechanical. Suppose you can write your equation so the highest time-derivative is isolated, u solved for in terms of lower derivatives, and the data surface is non-characteristic — meaning the equation actually lets you compute that top derivative from the data. Then the Cauchy data hands you u and its lower normal derivatives on the surface; the equation hands you the next normal derivative; differentiating the equation hands you the one after that; and so on, so you can read off every Taylor coefficient of u at a point of the surface. The theorem's real content is that when everything is analytic this formally generated power series actually converges in a neighbourhood, giving a genuine analytic solution. The non-characteristic condition is essential — on a characteristic surface the top derivative cannot be solved for, and the construction stalls.
Powerful as it is, the theorem comes with two honest limits that are central to the whole subject. First, it requires analyticity — a brutally strong assumption that excludes most realistic data (a temperature profile with a corner is not analytic), so it is far from the last word on existence. Second, and more strikingly, existence is not the same as well-posedness: Hadamard's example shows that the Cauchy problem for Laplace's equation has a solution by Cauchy-Kovalevskaya for analytic data, yet that solution depends discontinuously on the data and is therefore ill-posed. The theorem guarantees a local solution exists; it says nothing about whether the problem is stable, and for elliptic equations it is not.
For the wave equation u_tt = c^2 u_xx with analytic Cauchy data u(x,0)=f(x), u_t(x,0)=g(x) on the non-characteristic line t=0, the equation gives u_tt at t=0 directly from f''; differentiating gives all higher t-derivatives, and the Taylor series in t converges — an analytic solution exists locally. But run the same machine on Laplacian u = 0: a solution still exists, yet it is unstable (Hadamard), showing existence alone is not enough.
Cauchy-Kovalevskaya = local existence for analytic data on a non-characteristic surface; it does not promise stability.
The theorem proves existence, never well-posedness: its conclusion holds for elliptic equations like Laplace's where the Cauchy problem is ill-posed, so a Cauchy-Kovalevskaya solution can still depend discontinuously on the data. Existence and stability are separate questions.