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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 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 (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 solutions.

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 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 (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 solutions.

虽然常微分方程解的存在唯一性问题用'''<font color="#ff8000">弗罗贝尼乌斯定理 Picard–Lindelöf Theorem</font>得到了令人满意的结果，但应用到偏微分方程上却不尽如意。'''<font color="#ff8000">柯西-科瓦列夫斯基定理 Cauchy–Kowalevski theorem</font>指出，对于任何在未知函数及其导数中系数是解析的偏微分方程，柯西问题有一个局部唯一的解析解。虽然这个结果似乎解决了解的存在性和唯一性问题，但是有一些线性偏微分方程的系数具有所有级数的导数(尽管这些导数不是解析的) ，但是根本没有解: 见 Lewy (1957)。即使偏微分方程的解存在且唯一，它仍然可能具有人们所不希望的性质。这些问题的数学研究通常是在更有力的弱解的背景下进行的。

尽管常微分方程解的存在性和唯一性用'''<font color="#ff8000">弗罗贝尼乌斯定理 Picard–Lindelöf Theorem</font>得到了令人满意的结果，但偏微分方程解的存在性和唯一性却远没有得到解决。'''<font color="#ff8000">柯西-科瓦列夫斯基定理 Cauchy–Kowalevski theorem</font>指出：对于任意系数在未知函数及其导数中解析的偏微分方程，柯西问题存在一个局部唯一的解析解。虽然这个结果似乎解决了解的存在性和唯一性问题，但是存在一些线性偏微分方程-其系数具有所有级数的导数(尽管这些导数不是解析的) ，但是根本没有解: 见 Lewy (1957)。即使偏微分方程的解存在且唯一，它仍然可能具有不可预料的性质。这些问题的数学研究通常是在更有力的弱解的背景下进行的。

 

 

An example of pathological behavior is the sequence (depending upon {{mvar|n}}) of [[Cauchy problem]]s for the [[Laplace equation]]

An example of pathological behavior is the sequence (depending upon {{mvar|n}}) of [[Cauchy problem]]s for the [[Laplace equation]]