Cauchy–Kowalevski theorem

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 = K^m and W = Kⁿ. Let A₁, ..., A_n−1 be analytic functions defined on some neighbourhood of (0, 0) in W × V 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

${\displaystyle \partial _{x_{n}}f=A_{1}(x,f)\partial _{x_{1}}f+\cdots +A_{n-1}(x,f)\partial _{x_{n-1}}f+b(x,f)}$

with initial condition

${\displaystyle f(x)=0}$

on the hypersurface

${\displaystyle x_{n}=0}$

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, ..., 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 W × V. In this case, the same result holds.

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 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.

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

${\displaystyle \partial _{t}^{k}h=F\left(x,t,\partial _{t}^{j}\,\partial _{x}^{\alpha }h\right),{\text{ where }}j<k{\text{ and }}|\alpha |+j\leq k,}$

with initial conditions

${\displaystyle \partial _{t}^{j}h(x,0)=f_{j}(x),\qquad 0\leq j<k,}$

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.

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 ${\displaystyle Ext^{1}}$.

• 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 17–58.
• 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, doi:10.1007/978-3-642-96750-4, 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) [1994], "Cauchy–Kovalevskaya theorem", Encyclopedia of Mathematics, EMS Press

• PlanetMath