On the deformations of the incompressible Euler equations (Q2384749)

The author studies Cauchy problems of several systems of equations. These systems are obtained by certain transformations from the system of incompressible Euler equations. Let \(v(x,t)=(v_1,v_2,\dots,v_n)\) and \(p(x,t)\) be the solution to the problem \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=(1+\varepsilon)\|\nabla v(t)\|_{L^\infty}\,v,\quad \text{div}\,v=0, \quad x\in \mathbb R^n,\\ &v(x,0)=v_0(x).\end{aligned}\tag{1} \] Then the following properties of the solution of problem (1) are established: the existence of a local in time solution, a blow-up criterion, actual time blow-up if \(\text{curl}\,v_0\neq 0\), and a relation to the Euler equations. For the problem \[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+\nabla p=-(1+\varepsilon)\|\nabla v(t)\|_{L^\infty}\,v,\quad \text{div}\,v=0, \quad x\in \mathbb R^n,\\ &v(x,0)=v_0(x)\end{aligned}\tag{2} \] the existence of a global in time regular solution is obtained.