Backward and non-local problems for the Rayleigh-Stokes equation

TL;DR

This paper investigates the existence, uniqueness, and stability of solutions for the Rayleigh-Stokes problem with non-local conditions, including backward and forward cases, using Fourier methods and smooth data assumptions.

Contribution

It introduces conditions under which the Rayleigh-Stokes problem with non-local boundary conditions is well-posed, extending understanding of backward and non-local problems.

Findings

Backward problem is ill-posed with rough data but stable with smooth data.

Non-local problem with $eta=1$ is well-posed and satisfies coercive inequalities.

Fourier method provides conditions for existence and uniqueness of solutions.

Abstract

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: $u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta$ is either zero or one. It is well known that if $\beta=0$, 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 $\beta=1$, then the corresponding non-local problem is well-posed and coercive type inequalities are valid.