One-dimensional Compressible Fluid Research Articles

The Riemann problem for pressureless compressible fluid system with time- and space-dependent external force

We concern with Riemann problem for an one-dimensional pressureless compressible fluid system with time- and space-dependent external force. Under the concept of α-solution to this model, we obtain the α-solutions to the Riemann problem and the criteria of appearance of each α-solution. It is observed that the expressions of these α-solutions do not depend on α. Besides, the vacuum solution arises although the initial values do not involve vacuum. Furthermore, the delta wave solution emerges for certain initial values, in which internal energy variable and density variable contain the Dirac measure. Finally, our technique can be applied to study the other singular solutions to systems of conservation laws with a source term.

Global existence and time behaviour of strong solution to a compressible non-Newtonian fluid

This paper studies the initial–boundary problem of a one-dimensional compressible non-Newtonian fluid with vacuum, whose power-law exponent is r∈(1,2). The global existence of strong solution is obtained by combining the local existence and a priori estimates, provided that H1-norm of the initial density is small enough and the initial data satisfies a compatibility condition. Moreover, the time behaviour of ‖ρut‖L2 and ‖ux‖Lr are obtained on time interval of existence. As a byproduct, a blow-up criterion in term of ‖uxx‖L2 for strong solution to the initial–boundary value problem of the compressible non-Newtonian fluid. Our result is an extension of the work of Yuan and Xu (2008), where the local strong solution in one-dimensional space is obtained. Our analysis is based on some new mathematical techniques.

Global Existence of Strong Solutions to the Cauchy Problem for a One-Dimensional Compressible Non-Newtonian Fluid

Global Existence of Strong Solutions to the Cauchy Problem for a One-Dimensional Compressible Non-Newtonian Fluid

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.

Global stability of rarefaction waves for the 1D compressible micropolar fluid model with density-dependent viscosity and microviscosity coefficients

This paper is concerned with the global existence and large time behavior of strong solutions to the Cauchy problem of one-dimensional compressible isentropic micropolar fluid model with density-dependent viscosity and microviscosity coefficients, where the far-fields of the initial data are prescribed to be different. The pressure p(ρ)=ργ and the viscosity coefficient μ(ρ)=ρα for some parameters α,γ∈R are considered. For the case when the corresponding Riemann problem of the resulting Euler equations admits two rarefaction waves solutions, it is shown that if the parameters α and γ satisfy some conditions and the initial data is sufficiently regular, without vacuum and mass concentrations, then the Cauchy problem of the one-dimensional compressible micropolar fluid model has a unique global strong nonvacuum solution, which tends to a superposition of these two rarefaction waves as time goes to infinity. This result holds for arbitrarily large initial perturbation and large-amplitudes rarefaction waves. Moreover, the exponential time decay rate of the microrotation velocity ω(t,x) under large initial perturbation is also derived. The proof is given by an elaborate energy method and the key ingredient is to deduce the uniform-in-time lower and upper bounds on the specific volume.

Asymptotic behavior of the one-dimensional compressible micropolar fluid model

<p style='text-indent:20px;'>In this paper, we study the large time behavior of the solution for one-dimensional compressible micropolar fluid model with large initial data. This model describes micro-rotational motions and spin inertia which is commonly used in the suspensions, animal blood, and liquid crystal. We get the uniform positive lower and upper bounds of the density and temperature independent of both space and time. In particular, we also obtain the asymptotic behavior of the micro-rotation velocity.</p>

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.

Nonlinear Stability of Rarefaction Waves for a Compressible Micropolar Fluid Model with Zero Heat Conductivity

In 2018, Duan [1] studied the case of zero heat conductivity for a one-dimensional compressible micropolar fluid model. Due to the absence of heat conductivity, it is quite difficult to close the energy estimates. He considered the far-field states of the initial data to be constants; that is, $$\mathop {\lim }\limits_{x \to \pm \infty } ({v_0},{u_0},{\omega _0},{\theta _0})(x) = (1,0,0,1)$$ . He proved that the solution tends asymptotically to those constants. In this article, under the same hypothesis that the heat conductivity is zero, we consider the far-field states of the initial data to be different constants — that is, $$\mathop {\lim }\limits_{x \to \pm \infty } ({v_0},{u_0},{\omega _0},{\theta _0})(x) = ({v_ \pm },{u_ \pm },0,{\theta _ \pm })$$ -and we prove that if both the initial perturbation and the strength of the rarefaction waves are assumed to be suitably small, the Cauchy problem admits a unique global solution that tends time — asymptotically toward the combination of two rarefaction waves from different families.

Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line

<p style='text-indent:20px;'>In this paper, we study the asymptotic behavior of solutions to the initial boundary value problem for the one-dimensional compressible isentropic micropolar fluid model in a half line <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}_{+}: = (0, \infty). $\end{document}</tex-math></inline-formula> We mainly investigate the unique existence, the asymptotic stability and convergence rates of stationary solutions to the outflow problem for this model. We obtain the convergence rates of global solutions towards corresponding stationary solutions if the initial perturbation belongs to the weighted Sobolev space. The proof is based on the weighted energy method by taking into account the effect of the microrotational velocity on the viscous compressible fluid.

Strict dissipativity of Cattaneo–Christov systems for compressible fluid flow

This work considers a compressible, heat-conducting fluid exhibiting thermal relaxation according to Christov’s constitutive heat transfer law (Christov 2009 Mech. Res. Commun. 36 481–6), which is of Cattaneo type. The resulting evolution equations are known as Cattaneo–Christov systems. In this contribution, it is shown that Cattaneo–Christov systems for one-dimensional compressible fluid flow are strictly dissipative in both the viscous and inviscid cases. The proof is based on the verification of a genuine coupling condition for hyperbolic-parabolic systems with viscous and relaxation effects combined as well as on showing the existence of compensating functions of the state variables in the sense of Shizuta and Kawashima (1985 Hokkaido Math. J. 14 249–75). This property is used to obtain linear decay rates for solutions to the linearized equations around equilibrium states.

Concentration of mass in the pressureless limit of the Euler equations of one-dimensional compressible fluid flow

In this paper, we study the limiting behavior of Riemann solutions to the Euler equations of one-dimensional compressible fluid flow as γ tends to one. We show that the limit solution forms the delta wave to the pressureless Euler system of one-dimensional compressible fluid flow in the distribution sense. Some numerical results exhibiting the phenomena of concentration are also presented.

Global attractors for a nonlinear one-dimensional compressible viscous micropolar fluid model

This paper considers the dynamical behavior of solutions of constitutive systems for 1D compressible viscous and heat-conducting micropolar fluids. With proper constraints on initial data, we prove the existence of global attractors in generalized Sobolev spaces $$H_{\delta }^{(1)}$$ and $$H_{\delta }^{(2)}$$ . These attractors are unique in corresponding phase spaces.

Global stability of combination of a viscous contact wave with rarefaction waves for the compressible fluid models of Korteweg type

This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density- and temperature-dependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of a viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the capillarity coefficient satisfy some smallness conditions. The proof is based on some refined L2-energy estimates to control the possible growth of the solutions caused by the high nonlinearity of the system, the interactions of waves from different families and large data, and the key ingredient is to derive the uniform positive lower and upper bounds on the specific volume and the temperature.

Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model

In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L2-energy methods.

Asymptotic stability of a composite wave for the one-dimensional compressible micropolar fluid model without viscosity

We are concerned with the large time behavior of solutions to the Cauchy problem of the one-dimensional compressible micropolar fluid model without viscosity, where the far-field states of the initial data are prescribed to be different. If the corresponding Riemann problem of the compressible Euler system admits a contact discontinuity and two rarefaction waves solutions, we show that for such a non-viscous model, the combination of the viscous contact wave with two rarefaction waves is time-asymptotically stable provided that the strength of the composite wave and the initial perturbation are sufficiently small. The proof is given by an elementary L2 energy method.

Global classical solution to a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum

Because of the stronger nonlinearity and degeneracies of the density due to the momentum equation and appearance of vacuum, there are few results until now about global existence of classical solution to a compressible non-Newtonian fluid. In this paper, the global-in-time existence and uniqueness of classical solution to the initial–boundary-value problem is established for a one-dimensional compressible non-Newtonian fluid with large initial data and vacuum. Our analysis is based on some delicate energy estimates together with the iterate method being used ingeniously.

Global solutions for a one-dimensional compressible micropolar fluid model with zero heat conductivity

In this study, we consider global solutions for the one-dimensional compressible micropolar fluid model with zero heat conductivity, which is a hyperbolic–parabolic system. The pressure, velocity, and angular velocity are dissipative because of the viscosity, whereas the entropy is non-dissipative due to the absence of heat conductivity. Compared with the classical Navier–Stokes equations, there is an extra angular velocity ω in the micropolar fluid model, which causes both benefits and problems. The benefit is due to the fact that the term −vω is a damping term and it provides extra regularity for ω, and the problem is due to the term vω2 because it increases the nonlinearity of the system. The global solutions can be obtained by combining the local existence and a priori estimates if the H2-norm of the initial perturbation around a constant state is sufficiently small. The asymptotic behavior is also determined in this study.

Global strong solution for initial–boundary value problem of one-dimensional compressible micropolar fluids with density dependent viscosity and temperature dependent heat conductivity

In this paper, we prove existence and uniqueness of global strong solution of the compressible micropolar fluids model in one dimensional space with density dependent viscosity and temperature dependent heat conductivity under stress-free and thermally insulated boundary conditions. Former studies regard the coefficients as constants while we consider that the viscosity depends on density and the heat conductivity depends on temperature, which lead to the high nonlinearity and difficulty on deducing the bounds of both density and temperature.

Dissipation Limit for the Compressible Navier-Stokes to Euler Equations in One-Dimensional Domains

We prove that as the viscosity and heat-conductivity coefficients tend to zero, respectively, the global solution of the Navier-Stokes equations for one-dimensional compressible heat-conducting fluids with centered rarefaction data of small strength converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly away from the initial discontinuity.

Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type has been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient $\mu(\rho)=\rho^\alpha$, the capillarity coefficient $\kappa(\rho)=\rho^\beta$, and the heat-conductivity coefficient $\tilde{\alpha}(\theta)=\theta^\lambda$ for some parameters $\alpha,\beta,\lambda\in \mathbb{R}$ is considered. Under some assumptions on $\alpha,\beta$ and $\lambda$, we prove the global existence and time-asymptotic behavior of large solutions around constant states. The proofs are given by the elementary energy method combined with the technique developed by Y. Kanel' \cite{Y. Kanel} and the maximum principle.