Local existence with physical vacuum boundary condition to Euler equations with damping (Q1763961)

The initial boundary-value problem is considered for the equation \( (\mu^{-1}w_t)_t \) \(-(\mu w_y)_y +\) \(\mu^{-1}w_t =0\) in the domain \(t>0\), \(y>0\), with the boundary condition \(w| _{y=0}=0\), here \(\mu =(y^{-1}w)^{\alpha}\). The above problem results from the well-known \(p\)-system of gas dynamics in the Lagrangian variables. The term \(\mu^{-1}w_t\) is responsible for a linear damping. The boundary condition is derived under the assumption that the region \(y<0\) is the vacuum. It is proved that a smooth solution exists locally in time. The variable \(y\) is introduced through the density \(\rho\), the Euler variable \(x\), and the Lagrangian variable \(\xi\) by the formulas \(\xi=\)\(\int_o^x \rho (z,t)dz\), \(\xi=\) \(y^{2\gamma/(\gamma -1)}\), where the constant \(\gamma >1\) defines the pressure law \(p=\)\(\rho ^{\gamma}\).