Cauchy–Kovalevskaya 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 Kowalevski (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

 \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

 f(x) = 0

on the hypersurface

 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, ..., 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–Kowalevski theorem[edit]

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

 \partial_t^k h = F\left(x,t,\partial_t^j\,\partial_x^\alpha h \right),\text{ where }j<k\text{ and }|\alpha|+j\le k,\,

with initial conditions

 \partial_t^j h(x,0) = f_j(x),\qquad 0\le 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[edit]

The heat equation

 \partial_t h = \partial_x^2 h \,

with the condition

h(0,x) = {1\over 1+x^2}\text{ 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.)

The Cauchy–Kowalevski-Kashiwara theorem[edit]

There is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients, the Cauchy–Kowalevski-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 Ext^1.

Example[edit]

Let n\le m. Set Y=\{ x_1=\cdots=x_n \}. The system \partial_{x_i} f=g_i, i=1,\ldots,n, has a solution f\in \mathbb C \{ x_1,\ldots,x_m\} if and only if the compatibility conditions \partial_{x_i}g_j=\partial_{x_j}g_i are verified. In order to have a unique solution we must include an initial condition f|_Y=h, where h\in \mathbb C \{ x_{n+1},\ldots,x_m\}.

References[edit]

External links[edit]