更改

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

For well-posedness we require that the energy of the solution is non-increasing, i.e. that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>, which is achieved by specifying <math>u</math> at <math>x = a</math> if <math>\alpha > 0</math> and at <math>x = b</math> if <math>\alpha < 0</math>. This corresponds to only imposing boundary conditions at the inflow. Note that well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> holds when all data is set to zero.

对于适定性，我们要解的能量是不增加的，即 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> ，这种关系可以通过在<math>x = a</math>处 （如果 <math>\alpha > 0</math>） 以及 <math>x = b</math>处 （如果 <math>\alpha < 0</math>）指定<math>u</math>的值来实现。这只相当于在入流处附加边界条件。注意，适定性允许在数据(初始和边界)上的增长，因此它足以表明当所有数据设置为零时应有 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>。

对于适定性，我们要求解的能量是不增加的，即 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math> ，这种关系可以通过在<math>x = a</math>处 （如果 <math>\alpha > 0</math>） 以及 <math>x = b</math>处 （如果 <math>\alpha < 0</math>）指定<math>u</math>的值来实现。这只相当于在流入处附加边界条件。注意，适定性允许在数据项(初始和边界)上的增长，因此它足以表明当所有数据设置为零时应有 <math>\frac{\partial}{\partial t} \vert \vert u \vert \vert ^2 \leq 0</math>。

==Notes==

==Notes==