Well-posedness for the dumbbell model of polymeric fluids
Denote by \(u\) and \(p\) the velocity and pressure of the fluid, and \(Q\) the configuration of the spring, and hence the dumbbell, then \(Q\) obeys the following equation: \[ {\partial Q\over\partial t}+ (u\cdot\nabla) Q- (\nabla u)^TQ= -F(Q)+\dot w(t),\tag{1} \] where \(\dot w(t)\) is Gaussian white noise in time. The complete model for the full polymer-polymer system is then given by \[ {\partial u\over\partial t}+ (u\cdot\nabla)u+\nabla p= u+\nabla\tau,\quad \nabla u= 0,\tag{2} \] where \(\tau\) is the polymer contribution to stress, which is expressed via the Kramers expression \[ \tau(x,t)= E(F(Q)\otimes Q).\tag{3} \] The authors prove the following main result: Assume that the spring force \(F\) and the initial value satisfy the following conditions: (i) The function \(\gamma\) is \(C^\infty\) from \([0,+\infty)\) to \((0,+\infty)\), and \(\gamma'(| Q|^2)\geq 0\) and the derivative of \(F\) satisfies that \(|\nabla^m_{Q} F(Q)|\leq C(1+ | Q|^p)\) \((m= 0,1,2,3,4)\), where \(C\) is a constant and \(p\) is a certain nonnegative integer. (ii) \(\| u_0\|_{H^4}\leq \text{Const}.\), \(|\nabla^mQ_0|\leq \text{Const}.\) \((m= 0,1,2,3,4)\). Then there exists a \(T^*\) such that for \(t\leq T^*\), (1)--(3) has a unique strong solution \(u\) with \(u\in C^1([0, T^*], C^2(D))\cap L^2([0, T^*], H^5(D))\).
scientific article; zbMATH DE number 2114606