Polymer kinetic theory temperature dependent configurational probability diffusion equations: existence of positive solution results (Q6044587)

The authors consider an open domain \(\Omega \subset \mathbb{R}^{d}\), \(d=2\) or \(3\), with boundary \(\partial \Omega \in C^{1}\), the unit ball \(B(0,1)\), \(\Sigma =\Omega \times B(0,1)\), and \(\Sigma_{T}=\Sigma \times \lbrack 0,T)\), \(T>0\). The authors propose a configurational probability diffusion equation which accounts for temperature dependent molecular dynamics for an incompressible polymer fluid. The polymer chains are modeled as finitely extensible nonlinear elastic (FENE) dumbbells. The problem is stated as follows: find \(\psi :\Sigma_{T}\mapsto \mathbb{R}\), such that \(\frac{ \partial \psi }{\partial t}+\nabla_{x}\cdot (v\psi)-d_{1}\nabla_{x}\cdot \lbrack \nabla_{x}(\theta ln\psi)\psi ]+\nabla_{Q}\cdot (\kappa Q\psi)-d_{2}\nabla_{Q}\cdot (\theta \nabla_{Q}\psi)-d_{2}\delta \nabla_{Q}\cdot (\frac{2Q}{1-\left\Vert Q\right\Vert^{2}}\theta \psi)=0\), \(\forall (x,Q,t)\in \Sigma_{T}\), where \(\psi\) is the probability density, \(v\) the macroscopic velocity which satisfies \(\nabla_{x}\cdot v=0\), \(\theta\) the temperature, \(Q\) the end-to-end vector, \(\kappa =\nabla_{x}v\) the macroscopic velocity gradient, \(d_{1}\), \(d_{2}\) positive constants, and \(\delta >1\). The boundary conditions \([v\psi -d_{1}\nabla_{x}(\theta ln\psi)\psi ]\cdot \nu_{x}=0\), \(x\in \partial \Omega\), \(Q\in B(0,1)\), \(t\in (0,T)\), and \([\kappa Q\psi -d_{2}\theta \nabla_{Q}\psi -d_{2}\delta \frac{2Q}{ 1-\left\Vert Q\right\Vert^{2}}\theta \psi ]\cdot \nu_{Q}=0\), \(x\in \Omega\), \(Q\in \partial B(0,1)\), \(t\in (0,T)\) are imposed, together with the initial condition: \(\psi (x,Q,t=0)=\psi_{0}(x,Q)\), with for a given function \(\psi_{0}:\Sigma \rightarrow \mathbb{R}\). The authors write the corresponding variational formulation in the Hilbert space \(V=\{\phi :\Sigma \rightarrow \mathbb{R}:\int_{\Sigma }[\frac{1}{M}\phi^{2}+\frac{1}{M} \left\vert \nabla_{x}\phi \right\vert^{2}+M\left\vert \nabla_{Q}(\frac{ \phi }{M})\right\vert^{2}dxdQ<\infty \}\), where \(M(Q)=(1-\left\Vert Q\right\Vert)^{\delta }\). For the proof of the existence of a solution to this problem, they replace the term \(ln\psi\) in the above configurational probability diffusion equation by a regularizing function \(g_{\epsilon }(\psi)\), with \(\epsilon >0\) small enough. They prove the existence of a (weak) solution to the regularized problem mainly using compactness arguments. They obtain uniform estimates on this weak solution which allow to pass to the limit when \(\epsilon \rightarrow 0\) and they prove that the limit is a solution to the original variational formulation.