Stability of the rarefaction wave for a coupled compressible Navier‐Stokes/Allen‐Cahn system

In this paper, we are concerned with the large time behavior of solutions to the Cauchy problem for the one dimensional Navier‐Stokes/Allen‐Cahn system. Motivated by the relationship between the Navier‐Stokes/Allen‐Cahn system and the Navier‐Stokes system, we can prove that the solutions to the one‐dimensional compressible Navier‐Stokes/Allen‐Cahn system tend time‐asymptotically to the rarefaction wave, where the strength of the rarefaction wave is not required to be small. The proof is mainly based on a basic energy method.     

A remark on free boundary problem of 1‐D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we prove the existence and uniqueness of the weak solution of the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity µ(ρ)=ρ^θ with θ∈(0, γ?2], γ>1. The initial data are a perturbation of a corresponding steady solution and continuously contact with vacuum on the free boundary. The obtained results apply for the one‐dimensional Siant–Venant model of shallow water and generalize ones in (Arch. Rational Mech. Anal. 2006; 182: 223–253). Copyright © 2009 John Wiley & Sons, Ltd.     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

A Liouville theorem for the compressible Navier‐Stokes equations

This paper is concerned with 3‐dimensional steady compressible Navier‐Stokes equations. A Liouville‐type theorem is proved when some suitable conditions are satisfied.     

Blow‐up of the smooth solutions to the compressible Navier–Stokes equations

In this paper, we will firstly extend the results about Jiu, Wang, and Xin (JDE, 2015, 259, 2981–3003). We prove that any smooth solution of compressible fluid will blow up without any restriction about the specific heat ratio γ. Then we prove the blow‐up of smooth solution of compressible Navier–Stokes equations in half space with Navier‐slip boundary. The main ideal is constructing the differential inequality. Copyright © 2017 John Wiley & Sons, Ltd.     

Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier‐slip condition. The Navier‐slip condition differs in many aspects from no‐slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax–Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier‐slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L^∞ norm of the deformation tensor D(u). Copyright © 2015 John Wiley & Sons, Ltd.     

Relative energy approach to a diffuse interface model of a compressible two‐phase flow

We propose a simple model for a two‐phase flow with a diffuse interface. The model couples the compressible Navier‐Stokes system governing the evolution of the fluid density and the velocity field with the Allen‐Cahn equation for the order parameter. We show that the model is thermodynamically consistent, in particular, a variant of the relative energy inequality holds. As a corollary, we show the weak‐strong uniqueness principle, meaning any weak solution coincides with the strong solution emanating from the same initial data on the life span of the latter. Such a result plays a crucial role in the analysis of the associated numerical schemes. Finally, we perform the low Mach number limit obtaining the standard incompressible model.     

Blowup of classical solutions to the pressureless Euler/isentropic Navier‐Stokes equations

In this paper, we will show the blowup of classical solutions to the Cauchy problem for the pressureless Euler/isentropic Navier‐Stokes equations in arbitrary dimensions under some restrictions on the initial data. Compared with the degenerate viscosities appeared in the recent work, we consider the constant viscosities, but we can remove the condition that the adiabatic exponent has a upper bound, which was a key constraint in the proof of the blow‐up result is based on the construction of some new differential inequalities.     

Discretization of the Navier‐Stokes equations with slip boundary condition

We propose and analyze a two‐level method of discretizing the nonlinear Navier‐Stokes equations with slip boundary condition. The slip boundary condition is appropriate for problems that involve free boundaries, flows past chemically reacting walls, and other examples where the usual no‐slip condition u = 0 is not valid. The two‐level algorithm consists of solving a small nonlinear system of equations on the coarse mesh and then using that solution to solve a larger linear system on the fine mesh. The two‐level method exploits the quadratic nonlinearity in the Navier‐Stokes equations. Our error estimates show that it has optimal order accuracy, provided that the best approximation to the true solution in the velocity and pressure spaces is bounded above by the data. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 26–42, 2001     

Strong solutions of 3D compressible Oldroyd‐B fluids

This paper is concerned with a compressible viscoelastic fluids of Oldroyd‐B type. We prove the existence of unique local strong solutions for all initial data satisfying some compatibility condition. Moreover, we establish a blow‐up criterion for the strong solution in terms of the norm of the density tensor ρ and the norm of the symmetric tensor of constraints τ. All the results hold for the initial density vanishing from below. Copyright © 2012 John Wiley & Sons, Ltd.     

Semi‐invariant solutions of the Navier‐Stokes equations

We present new exact solutions and reduced differential systems of the Navier‐Stokes equations of incompressible viscous fluid flow. We apply the method of semi‐invariant manifolds, introduced earlier as a modification of the Lie invariance method. We show that many known solutions of the Navier‐Stokes equations are, in fact, semi‐invariant and that the reduced differential systems we derive using semi‐invariant manifolds generalize previously obtained results that used ad hoc methods. Many of our semi‐invariant solutions solve decoupled systems in triangular form that are effectively linear. We also obtain several new reductions of Navier‐Stokes to a single nonlinear partial differential equation. In some cases, we can solve reduced systems and generate new analytic solutions of the Navier‐Stokes equations or find their approximations, and physical interpretation.     

Low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions

In this paper, we consider low‐order stabilized finite element methods for the unsteady Stokes/Navier‐Stokes equations with friction boundary conditions. The time discretization is based on the Euler implicit scheme, and the spatial discretization is based on the low‐order element (P₁−P₁ or P₁−P₀) for the approximation of the velocity and pressure. Moreover, some error estimates for the numerical solution of fully discrete stabilized finite element scheme are obtained. Finally, numerical experiments are performed to confirm our theoretical results.     

Asymptotic behavior of solutions of the compressible Navier‐Stokes equations in a cylinder under the slip boundary condition

The large time behavior of solutions to the compressible Navier‐Stokes equations around the motionless state is considered in a cylinder under the slip boundary condition. It is shown that if the initial data are sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one‐dimensional nonlinear diffusion waves and a diffusive rigid rotation.     

Existence of global weak solutions for Navier‐Stokes‐Poisson equations with quantum effect and convergence to incompressible Navier‐Stokes equations

In this paper, we consider a three dimensional quantum Navier‐Stokes‐Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier‐Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Uniform regularity and vanishing viscosity limit for the chemotaxis‐Navier‐Stokes system in a 3D bounded domain

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis‐Navier‐Stokes system with Navier boundary condition for velocity field and Neumann boundary condition for cell density and chemical concentration in a 3D bounded domain. It is shown that there exists a unique strong solution of the incompressible chemotaxis‐Navier‐Stokes system in a finite time interval, which is independent of the viscosity coefficient. Moreover, this solution is uniformly bounded in a conormal Sobolev space, which allows us to take the vanishing viscosity limit to obtain the incompressible chemotaxis‐Euler system.     

Asymptotic stability of superposition of stationary solutions and rarefaction waves for 1D Navier–Stokes/Allen–Cahn system

In this paper, we investigate the large time behavior of the solutions to the inflow problem for the one-dimensional Navier–Stokes/Allen–Cahn system in the half space. First, we assume that the space-asymptotic states $({\rho }_{+},u_{+},{\chi }_{+})$ and the boundary data $({\rho }_b,u_b,{\chi }_b)$ satisfy some conditions so that the time-asymptotic state of solutions for the inflow problem is a nonlinear wave which is the superposition of a stationary solution and a rarefaction wave. Then, we show the existence of the stationary solution by the center manifold theorem. Finally, we prove that the nonlinear wave is asymptotically stable when the initial data is a small perturbation of the nonlinear wave. The proof is mainly based on the energy method by taking into account the effect of the concentration χ and the complexity of nonlinear wave.     

Global solutions to Keller‐Segel‐Navier‐Stokes equations with a class of large initial data in critical Besov spaces

In this article, we consider the Cauchy problem to Keller‐Segel equations coupled to the incompressible Navier‐Stokes equations. Using the Fourier frequency localization and the Bony paraproduct decomposition, let u_F:=e^tΔu₀; we prove that there exist 2 positive constants σ₀ and C₀ such that if the gravitational potential and the initial data (u₀,n₀,c₀) satisfy for some p,q with and , then the global solutions can be established in critical Besov spaces.