Cauchy–Kowalevski theorem

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

First order Cauchy–Kovalevskaya theorem[edit]

This theorem is about the existence of solutions to a system of m differential equations in n dimensions when the coefficients are analytic functions. The theorem and its proof are valid for analytic functions of either real or complex variables.

Let K denote either the fields of real or complex numbers, and let V = Km and W = Kn. Let A1, ..., An−1 be analytic functions defined on some neighbourhood of (0, 0) in V × W and taking values in the m × m matrices, and let b be an analytic function with values in V defined on the same neighbourhood. Then there is a neighbourhood of 0 in W on which the quasilinear Cauchy problem

with initial condition

on the hypersurface

has a unique analytic solution ƒ : W → V near 0.

Lewy's example shows that the theorem is not valid for all smooth functions.

The theorem can also be stated in abstract (real or complex) vector spaces. Let V and W be finite-dimensional real or complex vector spaces, with n = dim W. Let A1, ..., An−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 × W. In this case, the same result holds.

Proof by analytic majorization[edit]

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 that uniquely determine the coefficients. The Taylor series coefficients of the Ai'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 Ai'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–Kovalevskaya theorem[edit]

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

with initial conditions

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[edit]

The heat equation

with the condition

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.)

The Cauchy–Kovalevskaya–Kashiwara theorem[edit]

There is a wide generalization of the Cauchy–Kovalevskaya theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kovalevskaya–Kashiwara theorem, due to Masaki Kashiwara (1983). This theorem involves a cohomological formulation, presented in the language of D-modules. The existence condition involves a compatibility condition among the non homogeneous parts of each equation and the vanishing of a derived functor .

Example[edit]

Let . Set . The system has a solution if and only if the compatibility conditions are verified. In order to have a unique solution we must include an initial condition , where .

References[edit]

  • Cauchy, Augustin (1842), "Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles", Comptes rendus, 15  Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–53.
  • Folland, Gerald B. (1995), Introduction to Partial Differential Equations, Princeton University Press, ISBN 0-691-04361-2 
  • Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., 256, Springer, ISBN 3-540-12104-8, MR 0717035  (linear case)
  • Kashiwara, M. (1983), Systems of microdifferential equations, Progress in Mathematics, 34, Birkhäuser, ISBN 0817631380 
  • von Kowalevsky, Sophie (1875), "Zur Theorie der partiellen Differentialgleichung", Journal für die reine und angewandte Mathematik, 80: 1–32  (German spelling of her surname used at that time.)
  • Nakhushev, A.M. (2001), "Cauchy–Kovalevskaya theorem", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 

External links[edit]