Search

In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the compressible Navier–Stokes equations for ideal reacting gases. The asymptotic stability of the constant equilibrium state with strictly positive constant density, temperature and the vanishing velocity, mass fraction of the reactant is established under suitable small initial perturbation in H N ( R 3 ) ( N ⩾ 3 ) . Precisely, we show the convergence of the density, velocity and temperature towards the corresponding equilibrium state with the optimal rate ( 1 + t ) − 3 4 in L 2-norm as well as the convergence of the mass fraction to the equilibrium state with the optimal rate e − ϕ ( 1 ) t ( 1 + t ) − 3 4 in L 2-norm. Furthermore, the optimal decay rates for the spatial-derivatives of the solution are also obtained. The proof is based on the time-weighted energy estimate and continuation argument.

Because of the singular form and low flexibility of the clothoid as a highway transition curve in highway horizontal alignment, the K-curve, whose ratio of the chord-tangent angle to the tangential angle at any point is fixed, was proposed in a previous work, including the corresponding parametric equations and basic characteristics, and its high flexibility and various shapes were proved. Here, the application and verification of the K-curve in highway alignment is further studied. The solution methods of the K-curve for different connections are given. Through theoretical calculations and specific real-world examples, the geometric positions and lateral forces of the K-curve and the clothoid under different connection conditions are compared. The results show that the difference between the K-curve and clothoid in the above two aspects under a straight line-to-circle connection is greater than that under a circle-to-circle connection as an egg curve, and although the lateral force coefficient of the K-curve is smaller than that of the clothoid, the K-curve suffers from rapid change in the lateral force coefficient and a higher lateral acceleration change rate. Finally, the application conditions of the K-curve for highway alignment design are given with consideration of the centrifugal acceleration change rate.

We consider the initial boundary value problem for the isentropic compressible Navier-Stokes equations with cylindrical symmetry. The existence of boundary layers is well-known when the shear viscosity vanishes. In this paper, we derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate of the vanishing shear viscosity limit without any smallness assumption on the initial and boundary data.

The boundary value problem for the three-dimensional steady viscous compressible Navier-Stokes/Allen-Cahn (NSAC) system is considered. We establish the existence of a weak solution for the adiabatic index γ > 2 with the phase function belonging to [ − 1 , 1 ] . Moreover, we give further analysis on the weak solution. More specifically, we prove that the steady NSAC system degenerates into Navier-Stokes system if the phase function is positive (negative) on Ω ‾ .

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier–Stokes/Allen–Cahn system which is a combination of the classical Navier–Stokes system with an Allen–Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of [Formula: see text] for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier–Stokes/Allen–Cahn system. The proof is mainly based on a basic energy method.

In this paper, we study the approximation system to the steady compressible Navier–Stokes equations. It is proved that the density has a positive lower bound. And we use the lower bound to study the distribution of the density for a family of elliptic equations. A further investigation gives a typical upper bound of the density, and its limit functions only fit the solutions with the velocity divergence free.

In this paper, we construct a global weak solution to the compressible Navier-Stokes/Allen-Cahn system with possibly large initial data. The result holds in three dimensions subject to the conditions that the initial energy is finite and that the adiabatic constant γ > 2 .

For radar observations, invariant target parameters extracted from the scattering matrix (SM) provide information about the target’s geometry and composition. By studying the invariant target parameters of a small sample of insects, it is shown that the two eigenvalues and the determinant of the Graves power matrix are strongly correlated with insect body length and mass. Therefore, two methods are proposed to estimate the body length and mass from the eigenvalues or the determinant. The two eigenvalues identify the maximum of the insect polarization pattern and the perpendicular to the maximum direction, while the determinant is the product of these two values. A sample of 207 insect specimens measured at X-band in the laboratory rigs is used to determine the relationships between SM parameters and body lengths and masses. The results show that the length and mass can be estimated with a good performance and without an initial classification stage. In addition, the use of the determinant of the Graves power matrix is shown to provide an improved, SM-based, method of determining insect orientation and to solve the 90° orientation-extraction error problem that arises at X-band with very large insects.

In this paper, we consider the initial boundary value problem for the one-dimensional Navier–Stokes equations for viscous compressible and heat-conducting fluids in a bounded domain with the Robin boundary condition on temperature. There are few results until now about global existence of regular solutions to the full Navier–Stokes equations with the Robin boundary condition on temperature. By the analysis of the effect of boundary dissipation, we derive the global existence of classical solution to the corresponding initial boundary value problem with large initial data and vacuum. This result could be viewed as the first one on the global well-posedness of classical solutions to the full Navier–Stokes equations in a bounded domain with the Robin boundary condition on temperature.

This is a continuation of the paper [16, Math. Models Methods Appl. Sci. 28 (2018) 337–386] on the study of the large time behavior of the weak solution to the free boundary problem for one-dimensional isentropic compressible Navier–Stokes equations with degenerate viscosity and vacuum. Under appropriate smallness conditions on the initial data (initial energy), we extend the results in [16, Math. Models Methods Appl. Sci. 28 (2018) 337–386] to the case γ > 1 and θ min ⁡ { 1 , γ − 1 2 } . Clearly, the optimal decay rate of the density function along with its behavior near the interfaces is studied. In the meanwhile, we obtain also sharper decay rates for the norms in terms of the velocity function. The proof is based on the standard line method. The key is to establish some new global-in-time weighted (both in time and space) estimates uniformly up to the vacuum boundary, which ensures the uniform convergence of the approximate solutions.

In this paper we consider the non-stationary 1-D flow of a compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. Since the strong nonlinearity and degeneracies of the equations due to the temperature equation and vanishing of density, there are a few results about global existence of classical solution to this model. In the paper, we obtain a global classical solution to the equations with large initial data and vacuum. Moreover, we get the uniqueness of the solution to this system without vacuum. The analysis is based on the assumption \begin{document}$ \kappa(\theta) = O(1+\theta^q) $\end{document} where \begin{document}$ q\geq0 $\end{document} and delicate energy estimates.

A novel prediction method for gear friction coefficients with or without lubrication is developed with a computational inverse technique. A surrogate model is introduced to describe the mapping between the friction coefficients and root stresses, according to an adaptive radial basis function. The convergence criterion for solving the friction coefficients is determined based on the calculated and measured root stresses. A nonzero friction coefficient at the pitch point, which is closely relative to the action of rolling friction, is detected using the proposed method. Results show that the gear friction coefficients decrease with an increase in rotation speed or a decline in applied torque. Moreover, the friction coefficients without lubrication are more than twice those with lubrication.

The aim of this work is to study the global existence and large-time behavior of solutions to the non-isothermal model of capillary compressible fluids derived by Dunn and Serrin (1985). It is proved that the three-dimensional non-isothermal compressible Navier–Stokes–Korteweg system admits a unique global classical solution, provided that the initial energy is suitably small. This result improves previous results obtained by Hattori and Li (1996), where the existence of global classical solutions is established for the initial data close to an equilibrium in some Sobolev space H s ( s ≥ 3 ) . Furthermore, the large time behavior of the solution is also investigated.

In this paper, an iterative method is proposed for finding the spectral radius of an irreducible nonnegative tensor based on diagonal similar tensors. The iterative method is convergent for irreducible nonnegative tensors and has an explicit linear convergence rate for generalized weakly positive tensors. Some numerical results are provided to illustrate the efficiency of the iterative method.

In this paper, we propose a new viscosity for a coupled compressible Navier–Stokes/Allen–Cahn system because it describes the motion of a gas in a flowing liquid. The viscosity depends on two different variables (the density and the unknown function in Allen–Cahn equations). We establish blow-up criterions for strong solutions to initial-boundary value problem. We also show that the strong solutions can be improved to classical solutions. Precisely, when the viscosity only depends on the density, we prove the global existence and uniqueness of the Navier–Stokes/Allen–Cahn equations.

This paper is concerned with the initial-boundary value problem on the full Euler-Poisson system for ions over a half line. We establish the existence of stationary solutions under the Bohm criterion similar to the isentropic case and further obtain the large time asymptotic stability of small-amplitude stationary solutions provided that the initial perturbation is sufficiently small in some weighted Sobolev spaces. Moreover, the convergence rate of the solution toward the stationary solution is obtained. The proof is based on the energy method. A key point is to capture the positivity of the temporal energy dissipation functional and boundary terms with suitable space weight functions either algebraic or exponential depending on whether or not the incoming far-field velocity is critical.

The two-phase flow models are commonly used in industrial applications, such as nuclear, power, chemical-process, oil-and-gas, cryogenics, bio-medical, micro-technology and so on. This is a survey paper on the study of compressible nonconservative two-fluid model, drift-flux model and viscous liquid-gas two-phase flow model. We give the research developments of these three two-phase flow models, respectively. In the last part, we give some open problems about the above models.

In this paper, we are concerned with the global well-posedness and decay rates of strong solutions for the three-dimensional compressible Oldroyd-B model. We prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in H 2 -framework. Moreover, if the initial data belong to L 1 , the convergence rate of the solutions in L p -norm with 2 ⩽ p ⩽ 6 and convergence rates of their spatial derivatives in L 2 -norm are obtained. It is noteworthy that the smallness restriction on the coupling constant ω is not necessary in this paper.

In this paper, we give upper bounds on Zt-spectral radius of a tensor A ( t = 1 , 2 ) , which extend the upper bounds of Brauer to tensors. Moreover, an upper bound on the Z1-spectral radius is proposed via modulus sum of the entries of certain dimension of A , which improves the upper bound given by Li et al. Numerical experiments are given to illustrate the utility of the upper bound.

This paper is concerned with spherically symmetric motions of non-isentropic viscous gaseous stars with self-gravitation. When the stationary entropy S ‾ ( x ) is spherically symmetric and satisfies a suitable smallness condition, the existence and properties of the stationary solutions are obtained for 6 5 γ 2 with weaker constraints upon S ‾ ( x ) compared with the one in [26] , where γ is the adiabatic exponent. The global existence of strong solutions capturing the physical vacuum singularity that the sound speed is C 1 2 -Holder continuous across the vacuum boundary to a simplified system for non-isentropic viscous flow with self-gravitation and the nonlinear asymptotic stability of the stationary solution are proved when 4 3 γ 2 with the detailed convergence rates, motivated by the results and analysis of the nonlinear asymptotic stability of Lane–Emden solutions for isentropic flows in [29] , [30] .

In this paper, we are concerned with the asymptotic behavior of solutions to the system of Euler equations with time-depending damping, in particular, include the constant coefficient damping. We rigorously prove that the solutions time-asymptotically converge to the diffusion wave whose profile is self-similar solution to the corresponding parabolic equation, which justifies Darcy's law. Compared with previous results about Euler equations with constant coefficient damping obtained by Hsiao and Liu (1992) [2] , and Nishihara (1996) [9] , we obtain a general result when the initial perturbation belongs to the same space, i.e. H 3 ( R ) × H 2 ( R ) . Our proof is based on the classical energy method.

The geometric nonlinear vibrations of pretensioned orthotropic membrane with four edges fixed, which is commonly applied in building membrane structure, are studied. The nonlinear partial differential governing equations are derived by von Karman’s large deflection theory and D’Alembert’s principle. Because of the strong nonlinearity of governing equations, the homotopy perturbation method (HPM) to solve them is applied. The approximate analytical solution of the vibration frequency and displacement function is obtained. In the computational example, the frequency, vibration mode and displacement as well as the time curve of each feature point are analyzed. It is proved that HPM is an effective, simple and high-precision method to solve the geometric nonlinear vibration problem of membrane structures. These results provide some valuable computational basis for the vibration control and dynamic design of building and other analogous membrane structures.

The original Keller-Segel system proposed in [ 23 ] remains poorly understood in many aspects due to the logarithmic singularity. As the chemical consumption rate is linear, the singular Keller-Segel model can be converted, via the Cole-Hopf transformation, into a system of viscous conservation laws without singularity. However the chemical diffusion rate parameter e now plays a dual role in the transformed system by acting as the coefficients of both diffusion and nonlinear convection. In this paper, we first consider the dynamics of the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions. By imposing appropriate conditions on the boundary data, we show that boundary layer profiles are present as e →0 and large-time profiles of solutions are determined by the boundary data. We employ weighted energy estimates with the "effective viscous flux" technique to establish the uniform-in- e estimates to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we develop a new idea by exploring the convexity of an entropy expansion to get the basic L 1 -estimate. We the obtain the corresponding results for the original Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed to interpret our analytical results and their implications.

In this paper, the large time behavior of the solution to the initial-boundary problems for the one-dimensional compressible gas–liquid drift-flux model with slip is studied. Under some suitable smallness conditions upon the initial data, the optimal pointwise upper and lower decay estimates on masses as well as the sharpest decay rates for the norms in terms of the velocity function are obtained. This result generalizes the one in [On the large time behavior of the compressible gas–liquid drift-flux model with slip, Math. Models Methods Appl. Sci. 25 (2015) 2175–2215] by Evje and Wen. The key of the proof is to derive some new global-in-time weighted estimates. Our method can also be easily adopted to the study on the large time behavior of the solution to the one-dimensional compressible Naiver–Stokes equations.

In this paper, we investigate the global well-posedness of classical solutions to three-dimensional Cauchy problem of the compressible Navier–Stokes type system with a Korteweg stress tensor under the condition that the initial energy is small. This result improves previous results obtained by Hattori–Li in [H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal. 25 (1994) 85–98; H. Hattori and D. Li. Global solutions of a high-dimensional system for Korteweg materials. J. Math. Anal. Appl. 198 (1996) 84–97.], where the existence of the classical solution is established for initial data close to an equilibrium in some Sobolev space [Formula: see text].

This paper analytically and experimentally investigates the random vibration problem of pretensioned rectangular membrane structures under heavy rainfall. An analytical model is firstly proposed. In this model, the heavy rainfall is modeled as a series of discretized pulse load whose amplitude follows the non-Gaussian distribution; then, the nonlinear governing equation and corresponding Fokker Planck Kolmogorov (FPK) equation of membrane structures are derived and solved by the perturbation method, considering both the geometrical nonlinearity of structure and the non-Gaussian type of load; consequently, the analytical solutions of the probability density function (PDF), mean and standard deviation of displacement response can be obtained. An artificial heavy rainfall experimental system is then developed. This system is used to (i) verify the analytical model and (ii) evaluate the effects of rainfall intensity, pretension level, and material properties. The results show that the displacement follows the significant non-Gaussian distribution and proves strong nonlinear characteristics with increasing rainfall intensity and pretension. The proposed analytical model can guide engineers to design membrane structures from the perspective of probability.

We study the large-time asymptotic behavior of solutions toward the weak rarefaction wave of Navier–Stokes equations coupling with the Maxwell equations under small perturbations of initial data and also under the dielectric constant suitably small. The result is proved based on elementary L 2 energy methods.

In this paper, we consider the three dimensional compressible nonisentropic fluid model of Korteweg type. We mainly present the vanishing capillarity limit of smooth solution to the initial value problem. Based on the uniform estimates with respect to the capillary coefficient κ , we show the unique smooth solutions of the three dimensional Korteweg model converges globally in time to the smooth solution of the three dimensional compressible nonisentropic Navier–Stokes system, as κ tends to zero. Also, we give the convergence rate estimates in H 2 -norm with respect to κ .

In this paper, we consider the Dirichlet problem of a one-velocity viscous drift-flux model. One of the phases is compressible, the other one is weakly compressible. Under weak assumptions on the initial data, which can be discontinuous and large as well as involve transition to pure single-phase points or regions, we show existence of global bounded weak solutions. One main ingredient is that we employ a decomposition of the pressure term appearing in the mixture momentum equation into two components, one for each of the two phases. This paves the way for deriving a basic energy equality. In particular, upper bounds on the masses are extracted from the estimates provided by the energy equality. By relying on weak compactness tools we obtain an existence result within the class of weak solutions. An essential novel aspect of this analysis, compared to previous works on the same model, is that the solution space is large enough to allow for transition to single-phase flow without any constraints. In particular, one of the phases can vanish in a point while the other phase can persist. The key to achieve this result, which represents a major step forward compared to previous results for this model, is that we do not rely on any higher-order (i.e. derivatives in space) estimates on the masses or pressure, only low-order estimates provided by the energy equality and the uniform upper bounds on the liquid and gas mass.

In this paper, we give two Brauer-type eigenvalue inclusion sets and some bounds on the spectral radius for tensors. As applications, some bounds on the spectral radius of uniform hypergraphs are presented.

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of a rarefaction wave and a viscous contact discontinuity wave of the non-isentropic Navier–Stokes–Poisson system with free boundary. We first construct the composite wave through the quasineutral Euler equations and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding initial-boundary value problem of the non-isentropic Navier–Stokes–Poisson system. Only the strength of the viscous contact wave is required to be small. However, the strength of the rarefaction wave can be arbitrarily large. In our analysis, the domain decomposition plays an important role in obtaining the zero-order energy estimates. By introducing this technique, we successfully overcome the difficulty caused by the critical terms involved with the linear term, which does not satisfy the quasineural assumption for the composite wave. Copyright © 2016 John Wiley & Sons, Ltd.

This paper is concerned with the study of nonlinear stability of superposition of boundary layer and rarefaction wave on the two-fluid Navier–Stokes–Poisson system in the half line R + = : ( 0 , + ∞ ) . On account of the quasineutral assumption and the absence of the electric field for the large time behavior, we successfully construct the boundary layer and rarefaction wave, and then we give the rigorous proofs of the stability theorems on the superposition of boundary layer and rarefaction wave under small perturbations for the corresponding initial boundary value problem of the Navier–Stokes–Poisson system, only provided the strength of boundary layer is small while the strength of rarefaction wave can be arbitrarily large. The complexity of nonlinear composite wave leads to many complicated terms in the course of establishing the a priori estimates. The proofs are given by an elementary L 2 energy method.

The orthotropichttp://mts.hindawi.com/update/) in our Manuscript Tracking System and after you have logged in click on the ORCID link at the top of the page. This link will take you to the ORCID website where you will be able to create an account for yourself. Once you have done so, your new ORCID will be saved in our Manuscript Tracking System automatically."?>membrane structures have been popular in architectural structures. However, because of its lightweight and small stiffness, large nonlinear deflection vibration may occur under impact load, which leads to structural failure. In this paper, the governing equations of the large deflection nonlinear damped vibration of orthotropic saddle membrane structures excited by hailstone impact load are proposed according to the von Kármán’s large deflection theory and solved by applying the Bubnov–Galerkin method and the method of KBM perturbation. The approximate theoretical solution of the frequency function and displacement function of the large deflection nonlinear damped vibration of saddle membrane structures with four edges fixed excited by hailstone impact was obtained. The analytical examples proved that the mode shape function (equation (43)) can be applied to calculate the single-order mode shapes and the total superposed mode shapes of the damped large nonlinear deflection vibration of orthotropic saddle membrane structures excited by hailstone impact load succinctly. In addition, we compare and analyze the results of vibration frequency, amplitude, time histories, and total displacement of membrane structures with different pretensions and arch-to-span ratios under the impact of differently sized hailstones. The correctness of the analytical theory is verified by comparing with the results of numerical simulation. According to the results of this paper, we put forward some suggestions for the vibration control and dynamic design of practical spatial membrane structures.

We study the large-time asymptotic behavior of solutions toward the rarefaction wave of the compressible non-isentropic Navier-Stokes equations coupling with Maxwell equations under some small perturbations of initial data and also under the assumption that the dielectric constant is bounded. For that, the dissipative structure of this hyperbolic-parabolic system is studied to include the effect of the electromagnetic field into the viscous fluid and turns out to be more complicated than that in the simpler compressible Navier-Stokes system. The proof of the main result is based on the elementary \begin{document}$ L^2 $\end{document} energy methods.

In this article, we are concerned with the stability of stationary solution for outflow problem on the Navier-Stokes-Poisson system. We obtain the unique existence and the asymptotic stability of stationary solution. Moreover, the convergence rate of solution towards stationary solution is obtained. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. The proof is based on the weighted energy method by taking into account the effect of the self-consistent electric field on the viscous compressible fluid.

This paper is concerned with the study of the nonlinear stability of the contact discontinuity of the Navier-Stokes-Poisson system with free boundary in the case where the electron background density satisfies an analogue of the Boltzmann relation. We especially allow that the electric potential can take distinct constant states at boundary. On account of the quasineutral assumption, we first construct a viscous contact wave through the quasineutral Euler equations, and then prove that such a non-trivial profile is time-asymptotically stable under small perturbations for the corresponding initial boundary value problem of the Navier-Stokes-Poisson system. The analysis is based on the techniques developed in \cite{DL} and an elementary $L^2$ energy method.

In this paper, we are concerned with a nonconservative viscous compressible generic two-fluid model in $\mathbb{R}^3$, which is commonly used in industrial applications. The decay rates of classical solutions are established. Precisely, for any integer $s\geq 3$, we show that the velocities converge to the equilibrium states at the $L^2$-rate $(1+t)^{-\frac{3}{4}}$, and the $k(\in [1, s-2])$ order spatial derivatives of velocities converge to zero at the $L^2$-rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ as the compressible Navier--Stokes system, Navier--Stokes--Korteweg system, etc., but the fraction densities converge to the equilibrium states at the $L^2$-rate $(1+t)^{-\frac{1}{4}}$, and the $k(\in [1, s-1])$ order spatial derivatives of the fraction densities converge to zero at the $L^2$-rate $(1+t)^{-\frac{1}{4}-\frac{k}{2}}$, which are slower than the $L^2$-rate $(1+t)^{-\frac{3}{4}}$ and $L^2$-rate $(1+t)^{-\frac{3}{4}-\frac{k}{2}}$ for the compressible Navier--Stokes system, Navier--Stokes--Korteweg s...

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of lk,s-singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,s-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s-singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.

This paper is concerned with the large-time behavior of solutions to the Cauchy problem on the two-fluid Euler–Maxwell system with dissipation when initial data are around a constant equilibrium state. The main goal is the rigorous justification of diffusion phenomena in fluid plasma at the linear level. Precisely, motivated by the classical Darcy's law for the nonconductive fluid, we first give a heuristic derivation of the asymptotic equations of the Euler–Maxwell system in large time. It turns out that both the density and the magnetic field tend time-asymptotically to the diffusion equations with diffusive coefficients explicitly determined by given physical parameters. Then, in terms of the Fourier energy method, we analyze the linear dissipative structure of the system, which implies the almost exponential time-decay property of solutions over the high-frequency domain. The key part of the paper is the spectral analysis of the linearized system, exactly capturing the diffusive feature of solutions over the low-frequency domain. Finally, under some conditions on initial data, we show the convergence of the densities and the magnetic field to the corresponding linear diffusion waves with the rate [Formula: see text] in L2-norm and also the convergence of the velocities and the electric field to the corresponding asymptotic profiles given in the sense of the generalized Darcy's law with the faster rate [Formula: see text] in L2-norm. Thus, this work can be also regarded as the mathematical proof of the Darcy's law in the context of collisional fluid plasma.

In this paper, we investigate the convergence rates of the solution to the incompressible inhomogeneous fluid models of Korteweg type in three-dimensional whole space. We prove that as the capillarity coefficient and viscosity coefficient vanish, the local-in-time solution of the Cauchy problem for this system converges to the solution of the incompressible inhomogeneous Euler equations. In the end, we obtain a continuation theorem for the local smooth solution.

In this paper we study an asymptotic behavior of a solution to the initial boundary value problem for a viscous liquid-gas two-phase flow model in a half line $R_+:=(0,\infty).$ Our idea mainly comes from [23] and [29] which describe an isothermal Navier-Stokes equation in a half line. We obtain the convergence rate of the time global solution towards corresponding stationary solution in Eulerian coordinates. Precisely, if an initial perturbation decays with the algebraic or the exponential rate in space, the solution converges to the corresponding stationary solution as time tends to infinity with the algebraic or the exponential rate in time. These theorems are proved by a weighted energy method.

In this paper we are concerned with the initial–boundary value problem of the compressible gas–liquid model with well-formation interaction and gravity. The asymptotic behavior of solutions to steady states is established. Also the time-decay rates of perturbed solutions in the sense of L ∞ norm are obtained under some suitable assumptions on the initial date, if γ > 1 (associated with pressure law of gas) and β ∈ ( 0 , γ 2 ] ∩ ( 0 , γ − α γ ) ∩ ( 0 , γ + α γ 3 ] where β characterizes the viscosity coefficient and α describes the mass decay rate at the boundary. A main purpose of this work is to clarify the role played by the well-reservoir interaction term. The analysis demonstrates that it is essential to take into account information about sign as well as size of the interaction term in order to obtain time-independent estimates when it operates in combination with gravity.

This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad \partial_t u + \partial_x p(v) =\displaystyle -\frac{\alpha}{(1+t)^\lambda} u, \end{equation} with null-Dirichlet boundary condition or null-Neumann boundary condition on $u$. We show that the corresponding initial-boundary value problem admits a unique global smooth solution which tends time-asymptotically to the nonlinear diffusion wave. Compared with the previous work about Euler equations with constant coefficient damping, studied by Nishihara and Yang (1999, J. Differential Equations, 156, 439-458), and Jiang and Zhu (2009, Discrete Contin. Dyn. Syst., 23, 887-918), we obtain a general result when the initial perturbation belongs to the same space. In addition, our main novelty lies in the facts that the cut-off points of the convergence rates are different from our previous result about the Cauchy problem. Our proof is based on the classical energy method and the analyses of the nonlinear diffusion wave.