On the equations of one-dimensional motion of compressible viscous fluids

Some Uniform Estimates and Large-Time Behavior of Solutions to One-Dimensional Compressible Navier–Stokes System in Unbounded Domains with Large Data

“…Therefore, our results greatly improve the previous ones due to [5,8,12,14,17,22,23] where some additional smallness conditions on the initial data are needed. [9,10] where he proved that the temperature is uniformly (in time) bounded from below and above locally in x and that global solutions are convergent locally in space as time goes to infinity.…”

“…It remains to estimate the last term on the right hand side of (2.21). In fact, standard calculations yield that for any ε > 0, 22) where in the fourth and last inequalities we have used (2.4) and (2.1) respectively. Putting (2.22) into (2.21) and choosing ε suitably small lead to We will derive some necessary uniform estimates on the spatial derivatives of the global generalized solution (v, u, θ) in the next lemma.…”

Global existence and asymptotic behavior for the compressible Navier–Stokes equations with a non‐autonomous external force and a heat source

“…Qin [19] established the existence and exponential stability of a C 0 -semigroup in the subspace of H i × H i × H i (i = 1, 2) for a viscous ideal gas in a bounded domain in R and in a bounded annular domain G n = {x ∈ R n |0<a<|x|<b} (n = 2, 3) in R n for a viscous spherically symmetric ideal gas. This result improved those in [14] for an ideal gas and in [5] for the viscous spherically symmetric ideal gas in G n . As it is known, the constitutive equations of a real gas are well approximated within moderate ranges of u and by the model of an ideal gas.…”

Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation

“…[3], [5], [7], [8], [11], [12], [13], [17], [18], [14], [15], [16], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [31], [33], [35] and the references therein. When the initial data (v 0 (x), u 0 (x), s 0 (x)) is a small perturbation of a nonvacuum constant state, i.e., v − = v + > 0, u − = u + , s − = s + , satisfactory results have been obtained; cf.…”