Incompressible And Compressible Navier-Stokes Equations

Navier-Stokes equations describe the motion of viscous fluids and have abroadapplication. They are enriched with mathematics and physics, and relate to manymany partial diferential equations, so they play a central role in the theory of PDE.Leray，Hopf，Nash，P.L. Lions，Feferman，Cafarelli, Nirenberg, Ladyzhenskaya,Serrin, Kato, L.D. Landau, Kolmogorov, etc have done exellent work in Navier-Stokesequations. Feferman’s open problem is one of the seven millennium prize problems.This thesis contains two subproblems:1. Backward and Forward Self-Similar Solutions to2D Incompressible Naiver-Stokes EquationsFinding self-similar solutions to Navier-Stokes equations is important since self-similar solutions are related to the blow-up and asymptotic stability of the generic so-lutions. So we investigate the existence and uniqueness of backward and forward self-similar solution to2D incompressible Navier-Stokes equations. we obtained the follow-ing results:(i)There exist no backward and forward self-similar solutions satisfying theglobal energy estimates.(ii)There exist no backward self-similar solutions satisfyingthe local energy estimate.(iii)There exist one family of forward self-similar solutionto2D incompressible Navier-Stokes equations satisfying the local energy estimate, theyare Lamb-Oseen’s vortices. These results are consistent with the regularity and inversecascade phenomenon of2D incompressible Navier-Stokes equations.2. Classical Solvability of Compressible Isentropic Navier-Stokes Equationswith or without Initial VacuumWe investigate the classical solvability of the initial and boundary problem for com-pressible Navier-Stokes equations. we prove that:(i)As long as the initial density doesn’tvanish and the initial data satisfy some regularity, the classical solution is unique andglobally exists in the Sobolev-Hilbert type functional spaces. Recently, Cho and Kimhave already obtained the local classical solvability.(ii)When the initial density vanish-es, if the initial data satisfy no compatible conditions, then the isentropic Navier-Stokesequations with constant viscosity is ill-posed. If the initial data satisfy some compati-ble conditions, many papers obtained the local classical and strong solvability. We givea conjecture that for a large set of initial data contain vacuum near the boundary, the classical solution blows up in the finite time.