Global Existence of Regular Solutions with Large Oscillations and Vacuum

Li, Jing; Xin, Zhouping

doi:10.1007/978-3-319-10151-4_58-1

Cited by 6 publications

(11 citation statements)

References 39 publications

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“…Taking It is time to derive a uniform (in time) upper bound for the density, which plays a crucial role in deriving all the higher order estimates and thus extending the local classical solution to be a global one. The method we are going to employ can be found in [29] [30]. Lemma 3.8 Let (ρ, u) be a smooth solution of (1.1)-(1.5) on Ω×(0, T ] satisfying (3.9) and ∇u 0 L 2 ≤ M , then there exists a positive constant ε as described in Theorem 1.1 depending only on µ, λ, γ, a, ρ, ρ ∞ , Ω and M such that,…”

Section: Proof By Lemma 33 It Indicates Thatmentioning

confidence: 99%

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Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

Cai¹,

Li²,

Lü³

2021

Preprint

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We are concerned with the global existence of classical solutions to the barotropic compressible Navier-Stokes equations with slip boundary condition in a threedimensional (3D) exterior domain. We demonstrate that the classical solutions exists globally in time provided that the initial total energy is suitably small. It is worth noting that the initial density is allowed to have large oscillations and contain vacuum states. For our purpose, some new techniques and methods are adopted to obtain necessary a priori estimates, especially the estimates on the boundary. Moveover, we also give the large-time behavior of the classical solutions what we have gotten.

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Section: Proof By Lemma 33 It Indicates Thatmentioning

confidence: 99%

“…This section is devoted to some necessary higher order estimates of the smooth solution (ρ, u) which will make sure that the local existence of classical solution can be extended globally in time. The methods used are mainly from [30]. For completeness, we still write it down in detail.…”

Section: A Priori Estimates (Ii): Higher Order Estimatesmentioning

confidence: 99%

Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

Cai¹,

Li²,

Lü³

2021

Preprint

Self Cite

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“…In this subsection, we derive the time-dependent higher order estimates, which are necessary for the global existence of classical solutions. The procedure is similar as that in [3,19,20], and we sketch it here for completeness. From now on, assume that the initial energy C 0 ≤ ε 6 , and the positive constant C may depend on T, µ, λ, ν, a, γ, ρ ∞ , ρ, Ω, M 1 , M 2 , ∇u 0 H 1 , ∇H 0 H 1 , ρ 0 − ρ ∞ W 2,q , P (ρ 0 ) − P ∞ W 2,q , g L 2 for q ∈ (3, 6) where g ∈ L 2 (Ω) is given by compatibility condition (1.10).…”

Section: Time-dependent Higher Order Estimatesmentioning

confidence: 99%

“…In this section, we are prepared to proof the main results of this paper. Based on the estimates in Section 3, we follows the procedure in [3,20] to give the sketch of the proof.…”

Section: Time-dependent Higher Order Estimatesmentioning

confidence: 99%

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in a 3D Exterior Domain

Chen¹,

Huang²,

Shi³

2021

Preprint

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In this paper we study the initial-boundary-value problem for the barotropic compressible magnetohydrodynamic system with slip boundary conditions in threedimensional exterior domain. We establish the global existence and uniqueness of classical solutions to the exterior domain problem with the regular initial data that are of small energy but possibly large oscillations with constant state as far field which could be either vacuum or nonvacuum. In particular, the initial density of such a classical solution is allowed to have large oscillations contain vacuum states. Moreover, the large-time behavior of the solution is also shown.

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“…We now comment on this paper. Our work is motivated by Li-Xin [18], Li-Liang [16] and Cai-Li [4]. Our research is based on the following three important observations.…”

Section: Introductionmentioning

confidence: 99%

Global Classical Solutions to the 3D Density-Dependent Viscosity Compressible Navier-Stokes Equations with Navier-Slip Boundary Condition in a Simply Connected Bounded Domain

Cao

2021

Preprint

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This paper concerns the global existence for classical solutions problem to the 3D Density-Dependent Viscosity barotropic compressible Navier-Stokes in Ω with slip boundary condition, where Ω is a simply connected bounded C ∞ domain in R 3 and its boundary only has a finite number of 2-dimensional connected components. By a series of a priori estimates, we show that the classical solution to the system exists globally in time under the assumption that the initial energy is suitably small. The initial density of such a classical solution is allowed to have large oscillations and contain vacuum states. We also adopt some new techniques and methods to obtain necessary a priori estimates, especially the boundary integral terms estimates. This is the first result concerning the global existence of classical solutions to the compressible Navier-Stokes equations with density containing vacuum initially and viscosity coefficients depending on density for general 3D bounded smooth domains.

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Global Existence of Regular Solutions with Large Oscillations and Vacuum

Cited by 6 publications

References 39 publications

Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in a 3D Exterior Domain

Global Classical Solutions to the 3D Density-Dependent Viscosity Compressible Navier-Stokes Equations with Navier-Slip Boundary Condition in a Simply Connected Bounded Domain

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