Local existence and uniqueness of heat conductive compressible Navier-Stokes equations in the presence of vacuum and without initial compatibility conditions

In this paper, we investigate the initial-boundary value problem to the heat
conductive compressible Navier-Stokes equations. Local existence and uniqueness
of strong solutions is established with any such initial data that the initial
density $\rho_0$, velocity $u_0$, and temperature $\theta_0$ satisfy $\rho_0\in
W^{1,q}$, with $q\in(3,6)$, $u_0\in H^1$, and $\sqrt{\rho_0}\theta_0\in L^2$.
The initial density is assumed to be only nonnegative and thus the initial
vacuum is allowed. In addition to the necessary regularity assumptions, we do
not require any initial compatibility conditions such as those proposed in (Y.
Cho and H. Kim, \emph{Existence results for viscous polytropic fluids with
vacuum}, J. Differential Equations {\bf 228} (2006), no.~2, 377--411.), which
although are widely used in many previous works but put some inconvenient
constraints on the initial data. Due to the weaker regularities of the initial
data and the absence of the initial compatibility conditions, leading to weaker
regularities of the solutions compared with those in the previous works, the
uniqueness of solutions obtained in the current paper does not follow from the
arguments used in the existing literatures. Our proof of the uniqueness of
solutions is based on the following new idea of two-stages argument: (i)
showing that the difference of two solutions (or part of their components) with
the same initial data is controlled by some power function of the time
variable; (ii) carrying out some singular-in-time weighted energy differential
inequalities fulfilling the structure of the Gr\"onwall inequality. The
existence is established in the Euler coordinates, while the uniqueness is
proved in the Lagrangian coordinates first and then transformed back to the
Euler coordinates.

Local existence and uniqueness of heat conductive compressible Navier-Stokes equations in the presence of vacuum and without initial compatibility conditions - 42Papers