Barotropic Compressible Navier-Stokes Equations Research Articles

Global classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition in 2D bounded domains

We study the barotropic compressible Navier–Stokes equations with Navier-type boundary condition in a two-dimensional simply connected bounded domain with C∞ boundary ∂Ω. By some new estimates on the boundary related to the Navier-type slip boundary condition, the classical solution to the initial-boundary-value problem of this system exists globally in time provided the initial energy is suitably small even if the density has large oscillations and contains vacuum states. Furthermore, we also prove that the oscillation of the density will grow unboundedly in the long run with an exponential rate provided vacuum (even a point) appears initially. As we known, this is the first result concerning the global existence of classical solutions to the compressible Navier–Stokes equations with Navier-type slip boundary condition and the density containing vacuum initially for general 2D bounded smooth domains.

On Liouville-type theorem for the stationary compressible Navier–Stokes equations in $ \mathbb{R}^{3} $

<abstract><p>In this paper, we study the Liouville-type theorem for the stationary barotropic compressible Navier–Stokes equations in $ \mathbb{R}^{3} $. Based on a fairly general framework of a kind of local mean oscillations integral and Morrey spaces, we prove that the velocity and the density of the flow are trivial without any integrability assumption on the gradient of the velocity.</p></abstract>

Refined Pointwise Estimates for Solutions to the 1D Barotropic Compressible Navier–Stokes Equations: An Application to the Long-Time Behavior of a Point Mass

We study the long-time behavior of a point mass moving in a one-dimensional viscous compressible fluid. The author previously showed that the velocity of the point mass V(t) satisfies a decay estimate $$V(t)=O(t^{-3/2})$$ (Koike in J. Differ. Equ. 271:356–413, 2021). However, whether this decay estimate is optimal or not was not completely understood. In this paper, we answer this question by giving a simple necessary and sufficient condition for the validity of a lower bound of the form $$C^{-1}t^{-3/2}\le |V(t)|$$ for large t ( $$C>1$$ is a constant independent of t). This is achieved by refining the previously obtained pointwise estimates of solutions. We introduce inter-diffusion waves that, together with the classical diffusion waves, give an improved approximation of the fluid behavior around the point mass; this then leads to a sharper understanding of the long-time behavior of the point mass.

Controlling conservation laws II: Compressible Navier–Stokes equations

We propose, study, and compute solutions to a class of optimal control problems for hyperbolic systems of conservation laws and their viscous regularization [17]. We take barotropic compressible Navier–Stokes equations (BNS) as a canonical example. We first apply the entropy–entropy flux–metric condition for BNS. We select an entropy function and rewrite BNS to a summation of flux and metric gradient of entropy. We then develop a metric variational problem for BNS, whose critical points form a primal-dual BNS system. We design a finite difference scheme for the variational system. The numerical approximations of conservation laws are implicit in time. We solve the variational problem with an algorithm inspired by the primal–dual hybrid gradient method. This includes a new method for solving implicit time approximations for conservation laws, which seems to be unconditionally stable. Several numerical examples are presented to demonstrate the effectiveness of the proposed algorithm.

The Lp-Lq decay estimate for the multidimensional compressible flow with free surface in the exterior domain

The aim of this paper is to develop the general Lp theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in RN (N≥3). By the spectral analysis, we obtain the classical Lp-Lq decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation.

Vanishing viscosity limit of the three-dimensional barotropic compressible Navier–Stokes equations with degenerate viscosities and far-field vacuum

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for barotropic compressible fluids in $\mathbb{R}^3$. When the viscosity coefficients obey a lower power-law of the density (i.e., $\rho^\delta$ with $0<\delta<1$), we identify a quasi-symmetric hyperbolic--singular elliptic coupled structure of the Navier-Stokes equations to control the behavior of the velocity of the fluids near the vacuum. Then this structure is employed to prove that there exists a unique regular solution to the corresponding Cauchy problem with arbitrarily large initial data and far-field vacuum, whose life span is uniformly positive in the vanishing viscosity limit. Some uniform estimates on both the local sound speed and the velocity in $H^3(\mathbb{R}^3)$ with respect to the viscosity coefficients are also obtained, which lead to the strong convergence of the regular solutions of the Navier-Stokes equations with finite mass and energy to the corresponding regular solutions of the Euler equations in $L^{\infty}([0, T]; H^{s}_{\rm loc}(\mathbb{R}^3))$ for any $s\in [2, 3)$. As a consequence, we show that, for both viscous and inviscid flows, it is impossible that the $L^\infty$ norm of any global regular solution with vacuum decays to zero asymptotically, as $t$ tends to infinity. Our framework developed here is applicable to the same problem for the other physical dimensions via some minor modifications.

Non-existence of global classical solutions to barotropic compressible Navier-Stokes equations with degenerate viscosity and vacuum

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscosity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper.

Exponential decay for Lions-Feireisl's weak solutions to the barotropic compressible Navier-Stokes equations in 3D bounded domains

Exponential decay for Lions-Feireisl's weak solutions to the barotropic compressible Navier-Stokes equations in 3D bounded domains

Optimal decay for the compressible Navier-Stokes equations without additional smallness assumptions

This work is concerned with the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in Rd(d≥2). A pure energy argument (independent of spectral analysis) has been developed in more general Lp framework, which enables us not only to obtain the optimal time-decay rates but also to remove the smallness of low frequencies of initial data that has been required in previous studies on this problem. The crucial part of the proof lies in the evolution of negative Besov norms at low frequencies, which mainly depends on non standardLp product estimates and some elaborate use of Sobolev embeddings and interpolations. Our results could also hold true in case of large highly oscillating initial velocities in comparison with classical efforts.

Long-time behavior of a point mass in a one-dimensional viscous compressible fluid and pointwise estimates of solutions

We consider the motion of a point mass in a one-dimensional viscous compressible barotropic fluid. The fluid–point mass system is governed by the barotropic compressible Navier–Stokes equations and Newton's equation of motion. Our main result concerns the long-time behavior of the fluid and the point mass. Pointwise convergence estimates of the volume ratio and the velocity of the fluid to their equilibrium values are shown, and as a corollary, it is proved that the velocity V(t) of the point mass satisfies a decay estimate |V(t)|=O(t−3/2). The rate −3/2 is optimal and is essentially related to the compressibility and the nonlinearity. The main tool used in the proof is the pointwise estimates of Green's function. It turns out that the understanding of the time-decay properties of the transmitted and reflected waves at the point mass plays an essential role in the proof.

The [formula omitted]-bounded operator families arising from the study of the barotropic compressible flows with free surface

In this paper, we study some model problem associated to the free boundary value problem of the barotropic compressible Navier-Stokes equations in general smooth domain with taking surface tension into account. To obtain the maximal Lp−Lq regularity property of the model problem, we prove the existence of R-bounded operator families of the resolvent problem via Weis' theory on operator valued Fourier multipliers.

On local strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with vacuum

We consider the local well-posedness of strong and classical solutions to the three-dimensional barotropic compressible Navier-Stokes equations with density containing vacuum initially. We first prove the local existence and uniqueness of the strong solutions, where the initial compatibility condition proposed by Cho et al. (2004), Cho and Kim (2006) and Choe and Kim (2003) is removed in a suitable sense. Then, the continuous dependence of strong solutions on the initial data is derived under an additional compatibility condition. Moreover, for the initial data satisfying some additional regularity and the compatibility condition, the strong solution is proved to be a classical one.

Analyticity of solutions to the barotropic compressible Navier-Stokes equations

In this paper, we establish analyticity of solutions to the barotropic compressible Navier-Stokes equations describing the motion of the density ρ and the velocity field u in R3. We assume that ρ0 is a small perturbation of 1 and (1−1/ρ0,u0) are analytic in Besov spaces with analyticity radius ω>0. We show that the corresponding solutions are analytic globally in time when (1−1/ρ0,u0) are sufficiently small. To do this, we introduce the exponential operator e(ω−θ(t))D acting on (1−1/ρ,u), where D is the differential operator whose Fourier symbol is given by |ξ|1=|ξ1|+|ξ2|+|ξ3| and θ(t) is chosen to satisfy θ(t)<ω globally in time.

A well-posedness result for viscous compressible fluids with only bounded density

We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in the $L^\infty$ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension $d\geq2$ and for general pressure laws.

Global well-posedness and large-time behavior of 1D compressible Navier-Stokes equations with density-depending viscosity and vacuum in unbounded domains

We consider the Cauchy problem for one-dimensional (1D) barotropic compressible Navier-Stokes equations with density-depending viscosity and large external forces. Under a general assumption on the density-depending viscosity, we prove that the Cauchy problem admits a unique global strong (classical) solution for the large initial data with vacuum. Moreover, the density is proved to be bounded from above time-independently. As a consequence, we obtain the large time behavior of the solution without external forces.

Compressible fluids and active potentials

We consider a class of one dimensional compressible systems with degenerate diffusion coefficients. We establish the fact that the solutions remain smooth as long as the diffusion coefficients do not vanish, and give local and global existence results. The models include the barotropic compressible Navier-Stokes equations, shallow water systems and the lubrication approximation of slender jets. In all these models the momentum equation is forced by the gradient of a solution-dependent potential: the active potential. The method of proof uses the Bresch-Desjardins entropy and the analysis of the evolution of the active potential.

A Low-Frequency Assumption for Optimal Time-Decay Estimates to the Compressible Navier–Stokes Equations

The global existence issue in critical spaces for compressible Navier–Stokes equations, was addressed by Danchin (Invent Math 141:579–614, 2000) and then developed by Charve and Danchin (Arch Rational Mech Anal 198:233–271, 2010), Chen et al. (Commun Pure Appl Math 63:1173–1224, 2010) and Haspot (Arch Rational Mech Anal 202:427–460, 2011) in more general Lp setting. The main aim of this paper is to exhibit (more precisely) time-decay estimates of solutions constructed in the critical regularity framework. To the best of our knowledge, the low-frequency assumption usually plays a key role in the large-time asymptotics of solutions, which was firstly observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 1980) in the $${L^1(\mathbb{R}^d)}$$ space. We now claim a new low-frequency assumption for barotropic compressible Navier–Stokes equations, which may be of interest in the mathematical analysis of viscous fluids. Precisely, if the initial density and velocity additionally belong to some Besov space $${\dot{B}^{-\sigma_1}_{2,\infty}(\mathbb{R}^d)}$$ with the regularity $${\sigma_1\in (1-d/2, 2d/p-d/2]}$$ , then a sharp time-weighted inequality including enough time-decay information can be available, where optimal decay exponents for the high frequencies are exhibited. The proof mainly depends on some non standard Besov product estimates. As a by-product, those optimal time-decay rates of Lq–Lr type are also captured in the critical framework.

On scaling invariance and type-I singularities for the compressible Navier-Stokes equations

We find a new scaling invariance of the barotropic compressible Navier-Stokes equations. Then it is shown that type-I singularities of solutions with $$\mathop {\lim \sup }\limits_{t \nearrow T} |div(t,x)|(T - t) \leqslant \kappa $$ can never happen at time T for all adiabatic number γ > 1. Here κ > 0 does not depend on the initial data. This is achieved by proving the regularity of solutions under $$\rho (t,x) \leqslant \frac{M}{{{{(T - t)}^\kappa }}},M < \infty .$$ This new scaling invariance also motivates us to construct an explicit type-II blowup solution for γ > 1.

Free Boundary Value Problem for the Cylindrically Symmetric Compressible Navier-Stokes Equations with a Constant Exterior Pressure

This paper is concerned with the free boundary value problem (FBVP) for the cylindrically symmetric barotropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, the unique cylindrically symmetric strong solution is shown to exist globally in time and tend to a non-vacuum equilibrium state exponentially as time tends to infinity.

A Note on Time-Decay Estimates for the Compressible Navier–Stokes Equations

In the recent work, we have developed a decay framework in general L p critical spaces and established optimal time-decay estimates for barotropic compressible Navier–Stokes equations. Those decay rates of L q -L r type of the solution and its derivatives are available in the critical regularity framework, which were exactly firstly observed by Matsumura & Nishida, and subsequently generalized by Ponce for solutions with high Sobolev regularity. We would like to mention that our approach is likely to be effective for other hyperbolic/parabolic systems that are encountered in fluid mechanics or mathematical physics. In this paper, a new observation is involved in the high frequency, which enables us to improve decay exponents for the high frequencies of solutions.