We consider the Cauchy problem for the full compressible Navier-Stokes equations with vanishing of density at infinity in R 3 . Our main purpose is to prove the existence (and uniqueness) of global strong and classical solutions and study the large-time behavior of the solutions as well as the decay rates in time. Our main results show that the strong solution exists globally in time if the initial mass is small for the fixed coefficients of viscosity and heat conduction, and can be large for the large coefficients of viscosity and heat conduction. Moreover, large-time behavior and a surprisingly exponential decay rate of the strong solution are obtained. Finally, we show that the global strong solution can become classical if the initial data is more regular. Note that the assumptions on the initial density do not exclude that the initial density may vanish in a subset of R 3 and that it can be of a nontrivially compact support. To our knowledge, this paper contains the first result so far for the global existence of solutions to the full compressible NavierStokes equations when density vanishes at infinity (in space). In addition, the exponential decay rate of the strong solution is of independent interest.
In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique) for any T > 0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ, u) ∈ C 1 ([0, T ]; H 3 (R 1 )) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ 2 u tt , such as Lemmas 3.2-3.6. This leads to further regularities of (ρ, u)T ]; H 3 ([0, 1])). It is still open whether the regularity of u could be improved to C 1 ([0, T ]; H 3 ([0, 1])) with the appearance of vacuum, since it is not obvious that the solutions in C 1 ([0, T ]; H 3 ([0, 1])) to the initial boundary value problem must blow up in finite time.Crown
In the paper, we establish a blow-up criterion in terms of the integrability of the density for strong solutions to the Cauchy problem of compressible isentropic Navier-Stokes equations in R 3 with vacuum, under the assumptions on the coefficients of viscosity:This extends the corresponding results in [20,36] where a blow-up criterion in terms of the upper bound of the density was obtained under the condition 7µ > λ. As a byproduct, the restriction 7µ > λ in [12,37] is relaxed to 29µ 3 > λ for the full compressible Navier-Stokes equations by giving a new proof of Lemma 3.1. Besides, we get a blow-up criterion in terms of the upper bound of the density and the temperature for strong solutions to the Cauchy problem of the full compressible Navier-Stokes equations in R 3 . The appearance of vacuum could be allowed. This extends the corresponding results in [37] where a blow-up criterion in terms of the upper bound of (ρ, 1 ρ , θ) was obtained without vacuum. The effective viscous flux plays a very important role in the proofs.
In this paper, we are concerned with the Cauchy problem on the compressible isentropic two-fluids Euler-Maxwell equations in three dimensions. The global existence of solutions near constant steady states with the vanishing electromagnetic field is established, and also the time-decay rates of perturbed solutions in L q space for 2 ≤ q ≤ ∞ are obtained. The proof for existence is due to the classical energy method, and the investigation of the large-time behavior is based on the linearized analysis of the one-fluid Euler-Maxwell equations and the damped Euler equations. As a byproduct of our approach, some timedecay rates obtained in [18] for the nonlinear damped Euler system are improved.
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