A Quasilinear Parabolic Equation Describing the Elongation of Thin Filaments of Polymeric Liquids
Michael Renardy · SIAM Journal on Mathematical Analysis · 1982
We study the equation \[ \rho \ddot u = 3\eta \frac{{\partial ^2 }} {{\partial x\,\partial t}}\left( { - \frac{1} {{u_x }}} \right) + \frac{\partial } {{\partial x}}\int_{ - \infty }^t {a(t - s)} \left( {\frac{{u_x (t)}} {{u_x^2 (s)}} - \frac{{u_x (s)}} {{u_x^2 (t)}}} \right)ds \] where $u(x,t)$ is a real-valued function of $x \in [ - 1,1]$ and $t \in \mathbb{R}$, with the boundary condition \[3\eta \frac{\partial } {{\partial t}}\left( { - \frac{1}{{u_x }}} \right) + \int_{ - \infty }^t {a(t - s)} \left( {\frac{{u_x (t)}}{{u_x^2 (s)}} - \frac{{u_x (s)}} {{u_x^2 (t)}}} \right)ds = f(t)\] at $x = \pm 1$. This equation is derived as a model for the elongation of thin filaments of polymeric liquids, u denoting the position of a fluid particle in space, a the memory kernel, and f the force acting on the ends of the filament. We study the evolution of u, assuming the initial condition $u(x,t = - \infty ) = x$. It is shown that under appropriate conditions on a and f the boundary condition can be uniquely resolved with respect to $u_x $. The full problem is transformed in such a way that it is approachable by the Sobolevskii theory of quasilinear parabolic equations. This yields the existence of solutions to the initial value problem on sufficiently small time intervals. Moreover, we show that if $f(t)$ converges to zero exponentially as $t \to \pm \infty $ and is small in an appropriate norm, there exists a solution globally in time, which approaches a stationary limit as $t \to + \infty $.