Global solutions of a high dimensional system for Korteweg materials (Q1916768)

The main goal of this paper is to study the system for isothermal motion of Korteweg type: \[ \rho_t+(\rho u)_x+(\rho v)_y=0, \] \[ (\rho u)_t+(\rho u^2)_x+(\rho uv)_y= -\rho_x+\nu \rho\Delta\rho_x+ \mu\Delta u+(\mu/3)(u_{xx}+v_{xy}), \] \[ (\rho v)_t+(\rho uv)_x+ (\rho v^2)_y=-\rho_y+ \nu\rho\Delta\rho_y+ \mu\Delta v+(\mu/3)(u_{xy}+v_{yy}) \] with initial data \((\rho,u,v)(x,y,0)= (\rho_0,u_0,v_0)(x,y)\). The authors extend some known results, establish the global existence of the solution, and give necessary adjustments for the general high-dimensional case.