论文标题
雷利 - 斯托克斯方程的后退和非本地问题
Backward and non-local problems for the Rayleigh-Stokes equation
论文作者
论文摘要
傅里叶方法用于在右侧和雷利 - 斯托克斯问题中的初始数据上找到条件，从而确保解决方案的存在和唯一性。然后，在瑞利 - 斯托克斯问题中，而不是初始条件，请考虑非本地条件：$ u（x，t）=βU（x，0）+φ（x）$，其中$β$是零或一个。众所周知，如果$β= 0 $，则相应的问题（称为向后问题）在Hadamard的意义上是不适合的，即$ U（x，t）$的微小变化会导致初始数据的巨大变化。然而，我们将证明，如果我们考虑足够平滑的当前信息，那么解决方案就存在，并且它是独特而稳定的。还将表明，如果$β= 1 $，则相应的非本地问题是良好的，强制性的不等式是有效的。
The Fourier method is used to find conditions on the right-hand side and on the initial data in the Rayleigh-Stokes problem, which ensure the existence and uniqueness of the solution. Then, in the Rayleigh-Stokes problem, instead of the initial condition, consider the non-local condition: $u(x,T)=βu(x,0)+φ(x)$, where $β$ is either zero or one. It is well known that if $β=0$, then the corresponding problem, called the backward problem, is ill-posed in the sense of Hadamard, i.e. a small change in $u(x,T)$ leads to large changes in the initial data. Nevertheless, we will show that if we consider sufficiently smooth current information, then the solution exists and it is unique and stable. It will also be shown that if $β=1$, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.
