For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} where $S=\dfrac{\nabla v+\nabla v^T}{2}$ is the deformation rate tensor. The second-order differential operator $A(S,D)=[A_{ij}(S,D)]_{d\times d}$ is defined as follows with $d=2 \text{ or } 3$, and $A(S,D)$ is a strongly elliptic operator. $$ A_{ij}(S,D)\equiv \sum\limits_{l,k=1}^da_{ij}^{kl}(S)\partial_k\partial_l. $$ Actually, the incompressible Navier-Stokes system can be obtained by taking $a_{ij}^{kl}=\mu_0\delta_{ij}\delta_{kl}$ and $\mu_0$ is the Newtonian viscosity coefficient.

I want to investigate the Cauchy problem of the above system and seeking the local existence of its classical solution in $C([0,T_0],H^s)\cap C'([0,T_0],H^{s-2})$ for some positive constant $T_0$ with $s>d/2+2$ to be an integer.

I am able to manage the compressible case with the estimate in Sobolev space, however I have no idea in how to do it in the incompressible case. Could anyone help me figuring out how to prove the theorem? Or should I alter my existence theorem?

The long time blow-up or regularity of the solution is difficult for this system. Could anyone give some suggestions on the some results for the long-time behavior of the system I considered?

Thanks!

For the incompressible fluid dynamics system \begin{equation} \begin{split} &\nabla\cdot v=0,\\ &\dfrac{\partial v}{\partial t}+v\cdot \nabla v+\nabla p=A(S,D)v, \end{split} \end{equation} where $S=\dfrac{\nabla v+\nabla v^T}{2}$ is the deformation rate tensor. The second-order differential operator $A(S,D)=[A_{ij}(S,D)]_{d\times d}$ is defined as follows with $d=2 \text{ or } 3$, and $A(S,D)$ is a strongly elliptic operator. $$ A_{ij}(S,D)\equiv \sum\limits_{l,k=1}^da_{ij}^{kl}(S)\partial_k\partial_l. $$ Actually, the incompressible Navier-Stokes system can be obtained by taking $a_{ij}^{kl}=\mu_0\delta_{ij}\delta_{kl}$ and $\mu_0$ is the constant viscosity coefficient.

I am interested in the Cauchy problem of the above system and seeking the local existence of its classical solution in $C([0,T_0],H^s)\cap C'([0,T_0],H^{s-2})$ for some positive constant $T_0$ with $s>d/2+2$ to be an integer.

I am able to manage the compressible case with the estimate in Sobolev space, however I have no idea in how to do it in the incompressible case. Could anyone help me figuring out how to prove the theorem? Or should I alter my existence theorem?

The long time blow-up or regularity of the solution is difficult for this system. Could anyone give some suggestions on the some results for the long-time behavior of the system I considered?

Thanks!