Construction Of Blowup Solutions Of Two Types Of Fluid Equations

In this thesis,we study the construction of blowup solutions to fluid dynamics systems.Fluid dynamics concerns the dynamical phenomena of virous fluids in physics and engineering,such as liquid,gas,plasma,magnetic fluid and so on.There are compressible and incompressible fluid problems corresponding to compressible and incompressible Euler-Navier-Stokes equations.Besides,there are also systems coupled with other physical equations,which describe some specific fluids under some special conditions.For instance,Euler-Poisson system,Euler-Maxwell system and magnetohydrodynamics(MHD)system.One essential topic of the fluid dynamics research is whether a solution blows up in finite time with smooth data.The answer is positive in compressible Euler equation,while still unknown either in other compressible systems or incompressible problems.Recent years,many breakthroughs have been made in this area,such as works by Contantin,Majda,Kiselev,Sverak,Hou and Elgindi in incompressible fluid and works by Sideris,Christodoulou,Luk,Speck,Buckmaster,Shkoller and Vicol in compressible fluid.In this thesis,we mainly consider the Euler-NenerstPlanck-Poisson system and Euler-Poisson system with Ion background and construct blowup solutions.In the first of the thesis,we consider Euler-Nenerst-Planck-Poisson on R3 and show the existence of finite-time blowup solution with C1,α data.Here we exploit the framework built by Elgindi,which is used to constructed C1,α blowup solution to 3D incompressible Euler.Then we observe that the potential effect doesn’t dominate the system.Next we find a special solution to poential equation by using classical Bessel function theory and analyze the modulation equations as well as energy estimate to show the C1,α blowup solution to Euler is just the approximate blowup solution to ENPP.In the second part of the thesis,we consider the Euler-Poisson system with Ion background,which describes the behavior of plasma at low speed(not considering relativity effect).The existence of global solution with small initial data was obtained by Ionescu and Pausader for Euler-Maxwell system,so this thesis we mainly construct the blowup solutions with large initial data.Our argument is inspired by the works of Buckmaster Shkoller and Vicol,in which they constructed blowup solutions of 3D compressible Euler of large initial data.Here we substract 3D Burgers equation from the origin system and perform bootstrap argument to perturbation system.Finally we construct the blowup solutions to Euler-Poisson equation with large initial data by the esitmate of velocity field,modulation equations as well as higher order energy estimate.