Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations
The paper in question deals with compressible viscoelastic electrical conducting fluids. Their flows are governed by the elastic Navier-Stokes-Poisson system \begin{align*} \partial_t\rho + \nabla\cdot(\rho u) & = 0,\\ \partial_t(\rho u) + \nabla\cdot (\rho u \otimes u) + \nabla P(\rho) & = \mu\Delta u + (\mu + \lambda)\nabla\nabla\cdot u + c^2 \nabla\cdot(\rho \mathbb F\mathbb F^T) + \rho\nabla\Phi,\\ \partial_t\mathbb F + u \cdot \nabla \mathbb F & = \nabla u \mathbb F,\\ \Delta \Phi & = \rho - \overline\rho, \end{align*} which holds true on a certain bounded domain \(\Omega\subset \mathbb R^3\). Here the unknowns are the density \(\rho\), the velocity \(u,\) the deformation gradient \(\mathbb F,\) and the electrostatic potential \(\Phi\). The system is endowed with an initial condition \[(\rho, u, \mathbb F)(x,0) = (\rho_0,u_0,\mathbb F_0)(x),\] with the homogeneous Dirichlet bouundary condition for \(u\) and with either \[ \Phi|_{\partial\Omega} = 0\qquad \mbox{or }\quad \nabla\Phi\cdot \nu|_{\partial\Omega} = 0. \] The main theorem of this paper provides the global-in-time existence of a unique global solution. This result is obtained under additional assumption on the smallness of initial data \((\rho_0 - \overline\rho,u_0,\mathbb F_0)\). In order to obtain the necessary estimates allowing multiple usage of the local-in-time existence theorem, the authors work with the deformation \(\varphi\) rather than with the deformation gradient \(\mathbb F\). Here the deformation is defined as \(\varphi:= X(x,t) - x\), where \(X\) is the Lagrangian coordinate, i.e., an inverse to a function \(x(X,t)\) defined by the following ordinary differential equation \begin{align*} \frac{\mathrm{d}x(X,t)}{\mathrm{d}t} & = u(x(X,t),t),\\ x(X,0) & = X. \end{align*}
viscoelastic electrically conducting fluid
scientific article; zbMATH DE number 7379149
scientific article