Global existence of weak solutions for the Navier-Stokes equations with capillarity on the half-line (Q640386)

The authors we prove the existence of global weak solutions for the Navier-Stokes equations with capillarity: \[ v_t-u_x = 0, \qquad x\in [0,\infty), \qquad t>0, \tag{1} \] \[ u_t + p(v)_x = \left(\frac{u_x}{v}\right)_x-v_{xxx}, \qquad x\in[0,\infty), \qquad t>0, \tag{2} \] with Cauchy data \[ (v(x, 0), u(x, 0)) = (v_0(x), u_0(x)), \tag{3} \] such that \((v_0-v')\in H^1([0,\infty)),\) \(u_0 \in L^2([0,\infty))\), and boundary conditions \[ v_x(0, t) = u(0, t) = 0, \tag{4} \] where \(v\), \(u\), and \(p\) are the specific volume, velocity, and pressure in the fluid, \(v>0\) is a constant, and (1) and (2) are the equations of conservation of mass and balance of forces, respectively. It is assumed throughout that \[ p(v) = v^{-\gamma}, \qquad\gamma>1.\tag{5} \] The method of semi-discrete difference schemes, which is standard for Navier-Stokes equations in Lagrangian coordinates, is used to construct approximate solutions to the system (1)--(5). It is shown that these solutions satisfy various regularity estimates. Regular piecewise linear weak solutions are constructed.