更改

== Existence and uniqueness ==

== Existence and uniqueness ==

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the [[Picard–Lindelöf theorem]], that is far from the case for partial differential equations. The [[Cauchy–Kowalevski theorem]] states that the [[Cauchy problem]] for any partial differential equation whose coefficients are [[Analytic function|analytic]] in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see [[Lewy's example|Lewy (1957)]]. Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties.  The mathematical study of these questions is usually in the more powerful context of [[weak solution]]s.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the [[Picard–Lindelöf theorem]], that is far from the case for partial differential equations. The [[Cauchy–Kowalevski theorem]] states that the [[Cauchy problem]] for any partial differential equation whose coefficients are [[Analytic function|analytic]] in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see [[Lewy's example|Lewy (1957)]]. Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties.  The mathematical study of these questions is usually in the more powerful context of [[weak solution]]s.