Free boundary problem in a polymer solution model
The authors consider the integro-differential equation \[ \frac{\partial w}{\partial t}+\frac{\partial w}{\partial y}\int_{0}^{y}w(z,t)dz-w^{2}=\frac{\partial^{2}w}{\partial y^{2}}+\gamma (\frac{\partial^{3}w}{\partial y^{2}\partial t}+\frac{\partial^{3}w}{\partial y^{3}}\int_{0}^{y}w(z,t)dz- \frac{\partial^{2}w}{\partial y^{2}}), \] posed in the domain \(\Omega_{T} = \{y,t:0 < y < h(t)\), \(0\leq t\leq T\}\). This model accounts for the flow of a mixture of water and polymer. This equation is completed with: \(\frac{dh}{dt}=\int_{0}^{h}w(y,t)dy\). The boundary conditions \(w(0,t)=0\) and \(\frac{\partial w}{\partial y}+\gamma (\frac{\partial^{2}w}{\partial y\partial t}+ \frac{\partial^{2}w}{\partial y^{2}}\int_{0}^{h}w(y,t)dy-w\frac{\partial w}{\partial y})(h(t),t)=0\) are added, together with the initial conditions \( w(y,0)=w_{0}(y)\), \(0\leq y\leq 1\), \(h(0)=1\). Here \(\gamma >0\) is a constant and \(w_{0}\) is a smooth (\(C^{3}\)) function of \(y\) satisfying the conditions \( w_{0}(0)=w_{0}^{\prime}(1)=0\). The first main result proves the existence of a local in time strong solution (\(h\in C^{1}([0,t^{\ast}])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,t^{\ast}])\)) to this problem. If the initial condition further satisfies \(w_{0}(y)\leq 0\), \(w_{0}(y)-\gamma w_{0}^{\prime \prime}(y)\leq 0\), the authors prove the existence of a classical solution \(h\in C^{1}([0,T])\), \(w\in C^{3,1}([0,h(t)]\times \lbrack 0,T])\) to the above problem. Both existence results are obtained through appropriate transformations and using Schauder's theorem. The authors then consider the case where \(\gamma\) tends to 0 and they observe that the problem turns into that of the deformation of a strip of viscous fluid. They here prove that the solution to this problem is destructed in finite time. They finally introduce asymptotic expansions with respect to \(\gamma\) and they express the second term of this asymptotic expansion.
scientific article; zbMATH DE number 7344450