Large-Time Behavior of Solutions to the Multi-Dimensional Euler Equations with Damping

Abstract

We consider the three-dimensional compressible Euler equations of isentropic gases with damping:

$$ \begin{array}{*{20}{c}} {{{\rho }_{t}} + \nabla \cdot (\rho u) = 0,} \\ {\rho ({{u}_{t}} + u \cdot \nabla u) + \nabla P = - c\rho u,} \\ \end{array} $$

((1))

where ρx, t) ∈ ℝ, u(x, t) ∈ ℝ³ represent the density, velocity of the gas, respectively; x = (x ₁, x ₂, x ₃) ∈ ℝ³ is the space variable, t> 0 is the time variable; the pressure is P = P(ρ) = ρ^γ/γ with γ ≥ 1 as the adiabatic exponent; c > 0 is a constant, and ε = 1/c may be called the relaxation time for some physical flows. We study the Cauchy problem of (1) with the initial condition:

$$ (\rho ,u){{|}_{{t = 0}}} = ({{\rho }_{0}}(x),{{u}_{0}}(x)). $$

((2))

We are interested in the damping effect on the regularity and large-time behavior of the smooth solutions. It will be proved that the size of the smooth initial data in certain norms plays the key role. If the initial data are small in an appropriate norm, the damping can prevent the development of singularities in smooth solutions and the Cauchy problem has a unique global smooth solution. If the initial data are large in certain form, the damping is not strong enough to prevent the formation of singularities even though the initial data are smooth. This indicates that the smoothness of the solution will break down in a finite time.