Existence of nonstationary Poiseuille type solution under minimal regularity assumptions

时   间：  2023-04-04 14:00 — 15:00

The steady-state Poiseuille flow in an infinite straight pipe $\Pi=\{x: (x_1,x_2)\in \sigma, x_3\in (-\infty, +\infty)\}$ of constant cross-sectional $\sigma$ was described by Jean Louis Poiseuille in 1841. Today this classical solution of the Navier-Stokes equation seems to be trivial although it is used in numerous studies of fluid motion. The Poiseuille flow is characterised by the fact that the associated velocity field has only one nonzero component  $u(x)$ directed along the $x_n$-axis and it depends only on the variables $x^\prime$  of the cross-sectional $\sigma$, while the pressure function $p = p(x_n)$ is a linear function.

The Poiseuille-type solutions can be also defined in the non-steady case. Since in real applications one usually does not have data defined by smooth functions, it is important

to study non-stationary Poiseuille-type solutions assuming minimal regularity of data. This is the subject of the present talk. Existence and uniqueness of a very weak solution to the non-stationary Navier-Stokes equations having a prescribed flow rate (flux) in the infinite cylinder $\Pi$ are proved. It is assumed that the flow rate $F(t)$ is an element of $L^2(0, T)$ and the initial data $u_0 = (0, 0, u_{0n})$ is an element of $L^2(\sigma)$. The non-stationary Poiseuille solution has the form $u(x,t) = (0,0, U(x',t))$, $p(x, t) = - q(t)x(n) + p_0(t)$, where $(U(x', t), q(t))$ is a solution of an inverse problem for the heat equation with a specific over-determination condition. Under the above regularity assumptions the solution of the problem does not have the usual for parabolic problems regularity: it is much weaker.