Nontrivial equilibrium solutions and general 2 stability for stochastic evolution equations with pantograph delay and tempered fractional noise∗

Abstract

In this paper, we study the full compressible Navier--Stokes system in a bounded domain Ω⊂R3 , where the viscosity and heat conductivity depend on temperature in a power law (θb for some constant b>0 ) of Chapman--Enskog. We obtain the local existence of strong solution to the initial-boundary value problem (IBVP), which is not trivial, especially for the nonisentropic system with vacuum and temperature-dependent viscosity. There is degeneracy caused by vacuum, and there is extremely strong nonlinearity caused by variable coefficients, both of which create great difficulty for the a priori estimates, especially for the second-order estimates. First, in order to obtain closed first-order estimates, we introduce a new variable to reformulate the system into a better form and require the measure of initial vacuum domain to be sufficiently small. Second, with the help of a cut-off and straightening out technique, and the thermo-insulated boundary condition, we establish the time involved estimate for the second-order derivative of temperature, which plays a key role in closing the a priori estimates. Moreover, our local existence result holds for the cases that the viscosity and heat conductivity depend on θ with possibly different power laws (i.e., μ,λ∼θb1 , κ∼θb2 with constants b1,b2∈[0,+∞) ).