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第136行： − + 第144行：
第144行： − 病态行为的一个例子是拉普拉斯方程的 Cauchy 问题的序列(取决于)+ 第152行：
第152行： − 部分 x ^ 2} + 部分 y ^ 2}0，/ math 第168行：
第167行： − Math begin u (x，0) & 0， frac partial u }(x，0) & frac { sin n } ， end { align } / math 第176行：
第174行： − 哪里是整数。关于零的导数一致地随着增加而趋于零，但是解是+ 第184行：
第182行： − Math u (x，y) frac sinh ny sinn ^ 2} . / math 第192行：
第189行： − 如果不是任何非零值的整数倍，这个解决方案接近无穷大。拉普拉斯方程的 Cauchy 问题被称为不适定或不适定问题，因为该问题的解并不连续地依赖于该问题的数据。这种不适定问题在物理应用中通常不能令人满意。+ 第200行：
第197行： − 偏微分方程方程的解的存在性是千禧年大奖难题方程的一部分。+ − −
→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 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 Picard-lindel f 定理得到了令人满意的结果，但这与偏微分方程的情形相去甚远。柯西-科瓦列夫斯基定理指出，对于任何系数在未知函数及其导数中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，它仍然可能具有不希望的性质。这些问题的数学研究通常是在更强大的弱解的背景下进行的。
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，但这与偏微分方程的情形相去甚远。'''<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。
An example of pathological behavior is the sequence (depending upon ) of Cauchy problems for the Laplace equation
An example of pathological behavior is the sequence (depending upon ) of Cauchy problems for the Laplace equation
反常特征的一个例子是拉普拉斯方程的柯西问题的序列(取决于{{mvar|n}})
<math>\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0,</math>
<math>\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0,</math>
<math>\begin{align} u(x,0) &= 0, \\ \frac{\partial u}{\partial y}(x,0) &= \frac{\sin nx}{n}, \end{align}</math>
<math>\begin{align} u(x,0) &= 0, \\ \frac{\partial u}{\partial y}(x,0) &= \frac{\sin nx}{n}, \end{align}</math>
where is an integer. The derivative of with respect to approaches zero uniformly in as increases, but the solution is
where is an integer. The derivative of with respect to approaches zero uniformly in as increases, but the solution is
其中{{mvar|n}}是整数。{{mvar|u}}关于{{mvar|y}}的导数一致地随着{{mvar|n}}的增加而趋于零，但是解是
<math>u(x,y) = \frac{\sinh ny \sin nx}{n^2}.</math>
<math>u(x,y) = \frac{\sinh ny \sin nx}{n^2}.</math>
This solution approaches infinity if is not an integer multiple of for any non-zero value of . The Cauchy problem for the Laplace equation is called ill-posed or not well-posed, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.
This solution approaches infinity if is not an integer multiple of for any non-zero value of . The Cauchy problem for the Laplace equation is called ill-posed or not well-posed, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.
对于任何非零的{{mvar|y}}如果{{mvar|nx}}不是{{pi}}的整数倍，这个解会接近于无穷大。拉普拉斯方程的柯西问题被称为不适定问题（可以译为ill-posed或not well-posed），因为该问题的解并不连续地依赖于该问题的数据。这种不适定问题在物理应用中通常不能令人满意。
The existence of solutions for the Navier–Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems.
The existence of solutions for the Navier–Stokes equations, a partial differential equation, is part of one of the Millennium Prize Problems.
一个偏微分方程——纳维-斯托克斯方程——的解的存在性是千禧年大奖难题的一部分。
== Notation ==
== Notation ==