Fluid Models Of Korteweg Type Research Articles

Decay of higher order derivatives for Lp solutions to the compressible fluid model of Korteweg type

As a follow-up study of [32], we present a derivation of upper bounds for the decay of arbitrary higher order derivatives of solutions to the compressible Navier-Stokes-Korteweg system in the Lp critical Besov space. The approach, based on Gevrey regularizing estimates, is to establish uniform bounds on the growth of the radius of analyticity of the solution in negative Besov norms, which may be applicable to a wide range of dissipative systems.

Uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in half-space

In this paper, we study the uniform regularity and zero capillarity-viscosity limit for an inhomogeneous incompressible fluid model of Korteweg type in the half-space R+3 . We consider the Navier-slip boundary condition for velocity and the Dirichlet boundary condition for the gradient of density. By establishing the conormal energy estimates, we prove that there exists a unique strong solution of the model in a finite time interval [0,T0] , where T 0 is independent of the capillary and viscosity coefficients, and the solution is uniformly bounded in a conormal Sobolev space. Based on the aforementioned uniform estimates, we further show that there exists a constant 0<T1⩽T0 , such that the solutions of this model converge to the solution of the inhomogeneous incompressible Euler equations with the rates of convergence in L∞(0,T1;L2(R+3)) and L∞(0,T1;H1(R+3)) , as the capillary and viscosity coefficients tend to zero simultaneously.

Global existence and analyticity of Lp solutions to the compressible fluid model of Korteweg type

We are concerned with a system of equations in Rd(d≥3) governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed P′(ρ⁎)=0, it is found that the linearized system admits the purely parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of Lp-type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy ν¯2≥4κ¯, then the acoustic waves are not available in compressible fluids. Consequently, the prior L2 boundedness on the low frequencies of density and velocity could be improved to the general Lp version with 1≤p<d. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.

The global well-posedness of the compressible fluid model of Korteweg type for the critical case

In this paper, we consider the compressible fluid model of Korteweg type in a critical case where the derivative of pressure equals $0$ at a given constant state. We show that the system admits a unique, global strong solution for small initial data in the maximal $L_p$-$L_q$ regularity class. Consequently, we also prove the decay estimates of the solutions to the nonlinear problem. To obtain the global well-posedness for the critical case, we show $L_p$-$L_q$ decay properties of solutions to the linearized equations under an additional assumption for low frequencies.

Blow up of classical solutions to the barotropic compressible fluid models of Korteweg type in bounded domains

In this paper, we study the multi-dimensional barotropic compressible Navier–Stokes–Korteweg equations with degenerate coefficients of viscosities and capillary in bounded domains. It is shown that any classical solutions to the initial–boundary-value problem will blow up in finite time, when the initial density admits an isolated mass group. Moreover, a direct application of our result (μ(ρ)=μρ,λ=0,α=0) shows that the weak solution obtained by Bresch et al. (2003) cannot be improved to a classical one as long as there is an isolated mass group initially.

Asymptotic Behavior of Solutions to An Impermeable Wall Problem of the Compressible Fluid Models of Korteweg Type with Density-dependent Viscosity and Capillarity

This paper is concerned with the time-asymptotic behavior of strong solutions to the initial-boundary value problem of the isothermal compressible fluid models of Korteweg type with density-depende...

Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the compact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.

Existence of R‐bounded solution operator families for a compressible fluid model of Korteweg type on the half‐space

The aim of this paper is to show the existence of ‐bounded solution operator families for a generalized resolvent problem on the half‐space arising from a compressible fluid model of Korteweg type. Our main result covers not only the large resolvent parameter so that |λ|≥δ for some δ &gt; 0, but also small λ satisfying |λ|≤δ.

Lower bound of decay rate for higher-order derivatives of solution to the compressible fluid models of Korteweg type

This paper concerns the lower bound decay rate of global solution for compressible Navier–Stokes–Korteweg system in three-dimensional whole space under the H^{4}times H^{3} framework. At first, the lower bound of decay rate for the global solution converging to constant equilibrium state (1, 0) in L^2-norm is (1+t)^{-frac{3}{4}} if the initial data satisfy some low-frequency assumption additionally. Furthermore, we also show that the lower bound of the k(kin [1, 3])th-order spatial derivatives of solution converging to zero in L^2-norm is (1+t)^{-frac{3+2k}{4}}. Finally, it is proved that the lower bound of decay rate for the time derivatives of density and velocity converging to zero in L^2-norm is (1+t)^{-frac{5}{4}}.

Strong solutions for the compressible barotropic fluid model of Korteweg type in the bounded domain

This paper is concerned with a barotropic model of capillary compressible fluids derived by Dunn and Serrin (1985). We consider the initial boundary value problem in bounded domains of $${\mathbb {R}}^d (d=2,3)$$ with more general pressure including Van der Waals equation of state. First, the local in time existence and uniqueness of strong solutions is proved for initial data belonging to the Sobolev space $$W^{2}_{2}\times W^{1}_{2}$$, relying on a new linearization technique and the contraction mapping principle. Next, we show that there exists a global unique strong solution by the continuation argument of local solution under small initial perturbation. The proof is based on the elementary energy method, but with a new development, where some techniques are introduced to establish the uniform energy estimates and to treat the pressure function with non-increasing property.

Half-space model problem for a compressible fluid model of Korteweg type with slip boundary condition

In this paper, we consider a half-space model problem for a compressible fluid model of Korteweg type with slip boundary condition and prove the existence of ℛ-bounded solution operator families for the model problem.

Asymptotic profiles and convergence rate of the compressible fluid models of Korteweg type

In this paper, we consider the initial value problem for the compressible fluid models of Korteweg type in $${\mathbb {R}}^n(n\ge 3)$$ and asymptotic profile of global solutions and the corresponding convergence rate are established. The structure of the nonlinear term plays a very important role in constructing asymptotic profile. The proof is based on the decay estimate of solutions operator, decay estimate and weighted decay estimate of global solutions.

The Global Well-Posedness for the Compressible Fluid Model of Korteweg Type

In this paper, we consider the compressible fluid model of Korteweg type which can be used as a phase transition model. It is shown that the system admits a unique, global strong solution for small...

Stationary solutions to outflow problem for 1-D compressible fluid models of Korteweg type: Existence, stability and convergence rate

In this paper, we are concerned with the outflow problem in the half line (0,∞) to the isothermal compressible Navier–Stokes–Korteweg system with a nonlinear boundary condition for vanishing capillary tensor at x=0. We first give some necessary and sufficient conditions for the existence of the stationary solutions with the aid of center manifold theory. We also show the stability of the stationary solutions under smallness assumptions on the initial perturbation in the Sobolev space, by employing an energy method. Moreover, the convergence rate of the solution toward the stationary solutions is obtained, provided that the initial perturbations belong to the weighted Sobolev spaces.

On the maximal Lp-Lq regularity for a compressible fluid model of Korteweg type on general domains

The aim of this paper is to show the maximal Lp-Lq regularity for a compressible fluid model of Korteweg type on general domains of the N-dimensional Euclidean space for N≥2 (e.g. bounded domains; exterior domains; half-spaces, layers, tubes, and their perturbed domains). Our approach is based on the theory of the R-boundedness for a generalized resolvent problem associated with the Korteweg-type model.

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.

Compressible Fluid Model of Korteweg Type with Free Boundary Condition: Model Problem

The aim of this paper is to prove the existence of ${\mathcal R}$-bounded solution operator families for a resolvent problem on the upper half-space arising from a compressible fluid model of Korteweg type with free boundary condition. Such a compressible fluid model was derived by Dunn and Serrin (1985) and studied by Kotschote (2008) as a boundary value problem with non-slip boundary condition.

Strong solutions to the Cauchy problem of two-dimensional incompressible fluid models of Korteweg type

This paper studies the local existence of strong solutions to the Cauchy problem of the incompressible fluid models of Korteweg type with vacuum as far field density. The corresponding 3D problem has been solved by Tan and Wang (2010) [21]. Notice that the technique used by Tan and Wang fails treating the 2D case, because the Lp-norm (p>2) of the velocity u cannot be controlled in terms only of ρu and ∇u here. In the present paper, we will use the framework of weighted approximation estimates introduced by Liang (2015) [14] for Navier–Stokes equations to obtain the local existence of strong solutions provided the initial density does not decay very slowly at infinity, with the compact support case included.

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.

Unique solvability for the density-dependent incompressible Navier–Stokes–Korteweg system

This article concerns the strong solutions for the incompressible fluid models of Korteweg type with a density-dependent viscosity and capillary coefficient in a bounded domain Ω⊂R3. It is proved that the initial boundary problem of the density-dependent incompressible fluid of Korteweg type admits a unique local strong solution, which allows the existence of initial vacuum provided the initial data satisfies a compatibility condition.