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{{short description|Existence and uniqueness theorem for certain partial differential equations}}
{{Differential equations}}

In [[Mathematics|mathematics]], the '''Cauchy–Kovalevskaya theorem''' (also written as the '''Cauchy–Kowalevski theorem''') is the main local [[Existence theorem|existence]] and uniqueness theorem for [[Analytic function|analytic]] [[Partial differential equation|partial differential equation]]s associated with [[Cauchy problem|Cauchy initial value problem]]s. A special case was proven by {{harvs|txt=yes|last=Cauchy|first=Augustin|year=1842}}, and the full result by {{harvs|txt=yes|authorlink=Sofia Kovalevskaya|first=Sofya |last=Kovalevskaya|year=1874}}.

==First order Cauchy–Kovalevskaya theorem==
This theorem is about the existence of solutions to a system of ''m'' differential equations in ''n'' dimensions when the coefficients are [[Analytic function|analytic function]]s. The theorem and its proof are valid for analytic functions of either real or complex variables.

Let ''K'' denote either the [[Field (mathematics)|fields]] of real or complex numbers, and let ''V''&nbsp;=&nbsp;''K''<sup>''m''</sup> and ''W''&nbsp;=&nbsp;''K''<sup>''n''</sup>. Let ''A''<sub>1</sub>,&nbsp;...,&nbsp;''A''<sub>''n''−1</sub> be [[Analytic function|analytic function]]s defined on some [[Neighbourhood (mathematics)|neighbourhood]] of (0,&nbsp;0) in ''W''&nbsp;×&nbsp;''V'' and taking values in the ''m''&nbsp;×&nbsp;''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 [[Partial differential equation#Linear and nonlinear equations|quasilinear]] [[Cauchy problem]]

:<math> \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)</math>
with initial condition

:<math> f(x) = 0 </math>

on the hypersurface

:<math> x_n = 0 </math>

has a unique analytic solution ''ƒ''&nbsp;:&nbsp;''W''&nbsp;→&nbsp;''V'' near&nbsp;0.

[[Lewy's example]] shows that the theorem is not more generally 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''&nbsp;=&nbsp;dim&nbsp;''W''. Let ''A''<sub>1</sub>,&nbsp;...,&nbsp;''A''<sub>''n''−1</sub> be [[Analytic function|analytic function]]s with values in [[Endomorphism|End&nbsp;(''V'')]] and ''b'' an analytic function with values in ''V'', defined on some [[Neighbourhood (mathematics)|neighbourhood]] of (0,&nbsp;0) in ''W''&nbsp;×&nbsp;''V''. In this case, the same result holds.

==Proof by analytic majorization==
Both sides of the [[Partial differential equation|partial differential equation]] can be expanded as [[Formal power series|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''<sub>''i''</sub>'s and ''b'' are [[Majorization|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''<sub>''i''</sub>'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==
If ''F'' and ''f''<sub>''j''</sub> are analytic functions near 0, then the non-linear Cauchy problem

:<math> \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,</math>

with initial conditions

:<math> \partial_t^j h(x,0) = f_j(x),\qquad 0\le j<k,</math>

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|heat equation]]

:<math> \partial_t h = \partial_x^2 h </math>

with the condition

:<math>h(0,x) = {1\over 1+x^2}\text{ for }t = 0 </math>

has a unique formal power series solution (expanded around (0,&nbsp;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''|&nbsp;+&nbsp;''j''&nbsp;≤&nbsp;''k'' above cannot be dropped.  (This example is due to Kowalevski.)

==Cauchy–Kovalevskaya–Kashiwara theorem==

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
{{harvs|txt=yes|first=Masaki |last=Kashiwara|year=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|derived functor]] <math>Ext^1</math>.

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

==References==
*{{citation|last=Cauchy|first=Augustin|year=1842|title=Mémoire sur l'emploi du calcul des limites dans l'intégration des équations aux dérivées partielles|journal=Comptes rendus|volume=15|url=https://archive.org/stream/oeuvresdaugusti107caucrich#page/n27/mode/2up}} Reprinted in Oeuvres completes, 1 serie, Tome VII, pages 17–58.
*{{citation | last= Folland|first= Gerald B. | title= Introduction to Partial Differential Equations| year=1995| publisher= Princeton University Press| isbn = 0-691-04361-2|url=https://books.google.com/books?id=c0WFd3X_R20C&q=%22Cauchy%E2%80%93Kowalevski%22}}
*{{citation|mr=0717035|first=L.|last= Hörmander|title=The analysis of linear partial differential operators I|series= Grundl. Math. Wissenschaft. |volume= 256 |publisher= Springer |year=1983|isbn=3-540-12104-8 |doi=10.1007/978-3-642-96750-4}} (linear case)
*{{citation|first=M.|last= Kashiwara|title=Systems of microdifferential equations|series= Progress in Mathematics |volume= 34 |publisher= Birkhäuser |year=1983|isbn=0817631380  }}
*{{citation|last=von Kowalevsky|first= Sophie |journal = Journal für die reine und angewandte Mathematik|volume=80|year=1875|pages=1–32 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243919689_0080&DMDID=DMDLOG_0005|title=Zur Theorie der partiellen Differentialgleichung}} (German spelling of her surname used at that time.)
*{{springer|id=C/c020920|title=Cauchy–Kovalevskaya theorem|first=A.M.|last= Nakhushev}}

==External links==
*[https://planetmath.org/cauchykowalewskitheorem PlanetMath]

{{DEFAULTSORT:Cauchy-Kowalevski theorem}}
[[Category:Partial differential equations]]
[[Category:Theorems in analysis]]

{{Sourceattribution|Cauchy–Kowalevski theorem|1}}