The Blow-up Problem Of The3D Incompressible Navier-stokes Equations

Navier-Stokes equations is equations of motion which describes viscousincompressible fluid conservation of momentum conservation，depicts the basic lawsof mechanics of viscous fluid and has great significance in practical application. Thispaper studies the blow-up problem with the3D incompressible Navier-Stokesequations and mainly divided into the following two parts：The first part，we usually choose z-axis to the symmetry axis for the3Dsymmetry axis incompressible Navier-Stokes equations with swirl，in this case，theexpressions of evolution equations and vorticity are simplified. Assumingpressure on the axis of symmetry with local minimum value，this means that thez-axis has the lowest pressure， hence the pressure on the second order partialderivatives of radius is not less than zero on the axis，and here we put forwardblasting conditions of the solution in the finite time using the method of particletrajectory.The second part，we firstly establish a relationship between the properties ofthe vortex stretching and the geometric properties of the vorticity field by exploringthe geometric properties of the vorticity field along the vortex lines，then we obatinthe existence of blow-up solutions of three dimensional incompressible Navier-Stokesequations under certain conditions by applying the magnitude of vorticity equationand the famous differential geometry basic relationship.