Cauchy-Kowalevski theorem

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Categories: Partial differential equations

In mathematics, the Cauchy-Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. A special case was proved by Augustin Cauchy (1842), and the full result by Sophie Kowalevski (1875).

1 First order Cauchy-Kowalevski theorem
2 Proof by analytic majorization
3 Higher-order Cauchy-Kowalevski theorem
- 3.1 Example
4 References
5 External links

First order Cauchy-Kowalevski theorem

Let V and W be finite-dimensional vector spaces, with n = dim W. Let A₁, ..., A_n−1 be analytic functions with values in End (V) and b an analytic function with values in V, defined on some neighbourhood of (0,0) in V x W. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem

f _{x_n} = A₁(x,f) f_x₁ + ··· + A_n−1(x,f) f_{x_n−1} + b(x,f),

with initial condition

f(x) = 0 on x_n = 0,

has a unique analytic solution near 0.

The theorem and its proof are valid for analytic functions of either real or complex variables. Lewy's example shows that the theorem is not valid for smooth functions.

Proof by analytic majorization

Both sides of the partial differential equation can be expanded as formal power series and give recurrence relations for the coefficients of the formal power series for f which uniquely determine the coefficients. The Taylor series coefficients of the A_i's and b are majorized in matrix and vector norm by a simple scalar rational analytic function. The corresponding scalar Cauchy problem involving this function instead of the A_i's and b has an explicit local analytic solution. The absolute values of its coefficients majorize the norms of those of the original problem; so the formal power series solution must converge where the scalar solution converges.

Higher-order Cauchy-Kowalevski theorem

If F and f_j are analytic functions near 0, then the non-linear Cauchy problem

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_t^k h = F\left(x,t,\partial_t^j\,\partial_x^\alpha h \right)\,

with initial conditions

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_t^j h(x,0) = f_j(x)

where j < k and |α| + j ≤ k, has a unique analytic solution near 0.

This follows from the first order problem by considering the derivatives of h appearing on the right hand side as components of a vector-valued function.

Example

The heat equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \partial_t h = \partial_x^2 h

with the condition

Failed to parse (Missing texvc executable; please see math/README to configure.): h(0,x) = {1\over 1+x^2}

for t=0

has a unique formal power series solution (expanded around (0,0)). However this formal power series does not converge for any non-zero values of t, so there are no analytic solutions in a neighborhood of the origin. This shows that the condition |α| + j ≤ k above cannot be dropped. (This example is due to Kowalevski.)

References

Cauchy, Augustin (1842), Comptes Rend. 15 Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17-58.
Folland, Gerald B. (1995), Introduction to Partial Differential Equations, Princeton University Press, ISBN 0691043612
Hörmander, L. (1983), The analysis of linear partial differential operators I, vol. 256, Grundl. Math. Wissenschaft., Springer, MR 0717035, ISBN 3-540-12104-8 (linear case)
Kowalevski, Sophie (1875), "Zur Theorie der partiellen Differentialgleichung", Journal für die reine und angewandte Mathematik 80: 1-32, <http://docserver.digizeitschriften.de/digitools/resolveppn.php?PPN=509874> (The surname given in the paper is "von Kowalevsky", which may be a misprint.)
Nakhushev, A.M. (2001), "Cauchy-Kovalevskaya theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104