The initial value problem for the equation of motion of irrotational inviscid and heat conductive fluids

The Cauchy problem of the equation of motion of irrotational inviscid and heat conductive fluids is considered. It is proved that the heat diffusion prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution converges to their equilibrium state at the rate in the L2-norm as the non-isentropic compressible Navier–Stokes system.