Error estimates for two‐level penalty finite volume method for the stationary Navier–Stokes equations

Two‐level penalty finite volume method for the stationary Navier–Stokes equations based on the P₁ ? P₀ element is considered in this paper. The method involves solving one small penalty Navier–Stokes problem on a coarse mesh with mesh size H = ?^{1 / 4}h^{1 / 2}, a large penalty Stokes problem on a fine mesh with mesh size h, where 0 < ? < 1 is a penalty parameter. The method we study provides an approximate solution with the convergence rate of same order as the penalty finite volume solution (u_?h,p_?h), which involves solving one large penalty Navier–Stokes problem on a fine mesh with the same mesh size h. However, our method can save a large amount of computational time. Copyright © 2013 John Wiley & Sons, Ltd.     

On the global weak solutions for a modified two‐component Camassa‐Holm equation

In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})In this paper, we investigate the existence of global weak solutions to the Cauchy problem of a modified two‐component Camassa‐Holm equation with the initial data satisfying lim_{x → ±∞}u₀(x) = u_±. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution for the system under the assumption u_? ≤ u₊. The global weak solution is obtained as a limit of approximation solutions. The key elements in our analysis are the Helly theorem and the estimation of energy for approximation solutions in $H^1(\mathbb {R})\times H^1(\mathbb {R})$ and some a priori estimates on the first‐order derivatives of approximation solutions.     

The large time behavior of solutions to 3D Navier–Stokes equations with nonlinear damping

This paper studies the Cauchy problem of the 3D Navier–Stokes equations with nonlinear damping term | u | ^β?1u (β ≥ 1). For β ≥ 3, we derive a decay rate of the L²‐norm of the solutions. Then, the large time behavior is given by comparing the equation with the classic 3D Navier–Stokes equations. Copyright © 2011 John Wiley & Sons, Ltd.     

On the regularity criterion for the solutions of 3D Navier–Stokes equations in weak multiplier spaces

In this work, we improved the regularity criterion on the Cauchy problem for the Navier–Stokes equations in multiplier space in terms of the two partial derivatives of velocity fields, ?₁u₁ and ?₂u₂. Copyright © 2011 John Wiley & Sons, Ltd.     

Zero‐viscosity‐capillarity limit to rarefaction waves for the 1D compressible Navier–Stokes–Korteweg equations

In this paper, we study the zero viscosity and capillarity limit problem for the one‐dimensional compressible isentropic Navier–Stokes–Korteweg equations when the corresponding Euler equations have rarefaction wave solutions. In the case that either the effects of initial layer are ignored or the rarefaction waves are smooth, we prove that the solutions of the Navier–Stokes–Korteweg equation with centered rarefaction wave data exist for all time and converge to the centered rarefaction waves as the viscosity and capillarity number vanish, and we also obtain a rate of convergence, which is valid uniformly for all time. These results are showed by a scaling argument and elementary energy analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Remarks on classical solutions to steady quantum Navier-Stokes equations

The paper of Dong [Dong, J. Classical solutions to one-dimensional stationary quantum Navier–Stokes equations, J. Math Pure Appl. 2011] which proved the existence of classical solutions to one-dimensional steady quantum Navier–Stokes equations, when the nonzero boundary value u ₀ satisfies some conditions. In this paper, we obtain a different version of existence theorem without restriction to u ₀. As a byproduct, we get the existence result of classical solutions to the stationary quantum Navier–Stokes equations.     

Asymptotic behavior of solutions to anisotropic conservation laws in two-dimensional space

In this paper, we investigate the asymptotic behavior of solutions for anisotropic conservation laws in two-dimensional space, provided with step-like initial conditions that approach the constant states u_± (u₋<u₊) as x→±∞, respectively. It shows that there is a global classical solution that converges toward the rarefaction wave, ie, the unique entropy solution of the Riemann problem for the nonviscous Burgers' equation in one-dimensional space.     

Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in (0.1) $ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $ It is shown that two basic Riemann problems for Eq. (0.1) with the initial data $ S_ \mp (x) = \mp \operatorname{sgn} x $ exhibit a shock wave (u(x, t) ≡ S _?(x)) and a smooth rarefaction wave (for S ₊), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u _t + uu _x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ? ? ^N ) in the 1950s–1960s.     

Stability of the superposition of rarefaction wave and contact discontinuity for the non‐isentropic Navier–Stokes–Poisson system

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.     

Fisher–KPP equation with advection on the half‐line

We consider the Fisher–KPP equation with advection: u_t=u_xx?βu_x+f(u) on the half‐line x∈(0,∞), with no‐flux boundary condition u_x?βu = 0 at x = 0. We study the influence of the advection coefficient ?β on the long time behavior of the solutions. We show that for any compactly supported, nonnegative initial data, (i) when β∈(0,c₀), the solution converges locally uniformly to a strictly increasing positive stationary solution, (ii) when β∈[c₀,∞), the solution converges locally uniformly to 0, here c₀ is the minimal speed of the traveling waves of the classical Fisher–KPP equation. Moreover, (i) when β > 0, the asymptotic positions of the level sets on the right side of the solution are (β + c₀)t + o(t), and (ii) when β≥c₀, the asymptotic positions of the level sets on the left side are (β ? c₀)t + o(t). Copyright © 2016 John Wiley & Sons, Ltd.     

Remark on stability of rarefaction waves to the one‐dimensional compressible Navier–Stokes equations with density‐dependent viscosity coefficient

In this paper, we study one‐dimensional compressible isentropic Navier–Stokes equations with density‐dependent viscosity. We can obtain the asymptotic stability of rarefaction waves for the compressible isentropic Navier–Stokes equations when the power of viscosity coefficient , which enlarge the range of α in the article [Jiu Q, Wang Y, Xin ZP, Communication in Partial Differential Equations 2011; 36: 602‐634]. Copyright © 2013 John Wiley & Sons, Ltd.     

A Navier–Stokes–Voight model with memory

In this article, we consider a three‐dimensional Navier–Stokes–Voight model with memory where relaxation effects are described through a distributed delay. We prove the existence of uniform global attractors , where ? ∈ (0,1) is the scaling parameter in the memory kernel. Furthermore, we prove that the model converges to the classical three‐dimensional Navier–Stokes–Voight system in an appropriate sense as ? → 0. In particular, we construct a family of exponential attractors Ξ_? that is robust as ? → 0. Copyright © 2013 John Wiley & Sons, Ltd.     

The lower bounds of life span of classical solutions to one‐dimensional initial‐Neumann boundary value problems for general quasilinear wave equations

In this paper, we will study the lower bounds of the life span (the maximal existence time) of solutions to the initial‐boundary value problems with small initial data and zero Neumann boundary data on exterior domain for one‐dimensional general quasilinear wave equations u_tt?u_xx=b(u,Du)u_xx+F(u,Du). Our lower bounds of the life span of solutions in the general case and special case are shorter than that of the initial‐Dirichlet boundary value problem for one‐dimensional general quasilinear wave equations. We clarify that although the lower bounds in this paper are same as that in the case of Robin boundary conditions obtained in the earlier paper, however, the results in this paper are not the trivial generalization of that in the case of Robin boundary conditions because the fundamental Lemmas 2.4, 2.5, 2.6, and 2.7, that is, the priori estimates of solutions to initial‐boundary value problems with Neumann boundary conditions, are established differently, and then the specific estimates in this paper are different from that in the case of Robin boundary conditions. Another motivation for the author to write this paper is to show that the well‐posedness of problem 1.1 is the essential precondition of studying the lower bounds of life span of classical solutions to initial‐boundary value problems for general quasilinear wave equations. The lower bound estimates of life span of classical solutions to initial‐boundary value problems is consistent with the actual physical meaning. Finally, we obtain the sharpness on the lower bound of the life span 1.8 in the general case and 1.10 in the special case. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in L² norm. More precisely, the velocities u₁ and u₂ decay at the L²?rate , which is faster than the normal L²‐rate for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities ρ₁?ρ₂ and the disparity of two velocities u₁?u₂ could reach to and in L² norm, respectively. Copyright © 2015 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.     

Finite volume method based on stabilized finite elements for the nonstationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two‐dimensional nonstationary Navier–Stokes equations is investigated in this work. As in stabilized finite element method, macroelement condition is introduced for constructing the local stabilized formulation of the nonstationary Navier–Stokes equations. Moreover, for P₁ ? P₀ element, the H¹ error estimate of optimal order for finite volume solution (u_h,p_h) is analyzed. And, a uniform H¹ error estimate of optimal order for finite volume solution (u_h, p_h) is also obtained if the uniqueness condition is satisfied. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Leading term at infinity of steady Navier‐Stokes flow around a rotating obstacle

Consider a viscous incompressible flow around a body in $\mathbb R^3$ rotating with constant angular velocity ω. Using a coordinate system attached to the body, the problem is reduced to a modified Navier‐Stokes system in a fixed exterior domain. This paper addresses the question of the asymptotic behavior of stationary solutions to the new system as |x| → ∞. Under a suitable smallness assumption on the velocity field, u, and the net force on the boundary, N, we prove that the leading term of u is the so‐called Landau solution U, a singular solution of the stationary Navier‐Stokes system in $\mathbb R^3$ with external force kωδ₀ and decaying as 1/|x|; here $k\in \mathbb R$ is a suitable constant determined by N and δ₀ is the Dirac measure supported in the origin.     

An algebraic multigrid method for Q2−Q1 mixed discretizations of the Navier–Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily colocated at mesh points. Specifically, we investigate a Q ₂? Q ₁ mixed finite element discretization of the incompressible Navier–Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees of freedom (DOFs) are defined at spatial locations where there are no corresponding pressure DOFs. Thus, AMG approaches leveraging this colocated structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity DOF relationships of the Q ₂? Q ₁ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity DOFs resembles that on the finest grid. To define coefficients within the intergrid transfers, an energy minimization AMG (EMIN‐AMG) is utilized. EMIN‐AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier–Stokes problems.     

Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.     

Long‐time dynamics of KdV solitary waves over a variable bottom

We study the variable‐bottom, generalized Korteweg—de Vries (bKdV) equation ?_tu = ??_x(?u + f(u) ? b(t,x)u), where f is a nonlinearity and b is a small, bounded, and slowly varying function related to the varying depth of a channel of water. Many variable‐coefficient KdV‐type equations, including the variable‐coefficient, variable‐bottom KdV equation, can be rescaled into the bKdV. We study the long‐time behavior of solutions with initial conditions close to a stable, b = 0 solitary wave. We prove that for long time intervals, such solutions have the form of the solitary wave whose center and scale evolve according to a certain dynamical law involving the function b(t,x) plus an H¹(?)‐small fluctuation. © 2005 Wiley Periodicals, Inc.