Stokes and Navier–Stokes equations with nonhomogeneous boundary conditions

In this paper, we study the existence and regularity of solutions to the Stokes and Oseen equations with nonhomogeneous Dirichlet boundary conditions with low regularity. We consider boundary conditions for which the normal component is not equal to zero. We rewrite the Stokes and the Oseen equations in the form of a system of two equations. The first one is an evolution equation satisfied by Pu, the projection of the solution on the Stokes space – the space of divergence free vector fields with a normal trace equal to zero – and the second one is a quasi-stationary elliptic equation satisfied by (I−P)u, the projection of the solution on the orthogonal complement of the Stokes space. We establish optimal regularity results for Pu and (I−P)u. We also study the existence of weak solutions to the three-dimensional instationary Navier–Stokes equations for more regular data, but without any smallness assumption on the initial and boundary conditions.     

Well‐posedness for fractional Navier–Stokes equations in the largest critical spaces

This note studies the well‐posedness of the fractional Navier–Stokes equations in some supercritical Besov spaces as well as in the largest critical spaces for β ∈ (1/2,1). Meanwhile, the well‐posedness for fractional magnetohydrodynamics equations in these Besov spaces is also studied. Copyright © 2012 John Wiley & Sons, Ltd.     

Stabilized spectral element approximation for the Navier–Stokes equations

The conforming spectral element methods are applied to solve the linearized Navier–Stokes equations by the help of stabilization techniques like those applied for finite elements. The stability and convergence analysis is carried out and essential numerical results are presented demonstrating the high accuracy of the method as well as its robustness. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 115–141, 1998     

Moment estimates for weak solutions to the Navier–Stokes equations in half‐space

We establish the moment estimates for a class of global weak solutions to the Navier–Stokes equations in the half‐space. Copyright © 2009 John Wiley & Sons, Ltd.     

Strong solutions for the nonhomogeneous Navier–Stokes equations in unbounded domains

We show the existence of strong solutions for the nonhomogeneous Navier–Stokes equations in three‐dimensional domains with boundary uniformly of class C³. Under suitable assumptions, uniqueness is also proved. Copyright © 2009 John Wiley & Sons, Ltd.     

A note on integration of the Navier–Stokes equations

In this study the 2D Navier–Stokes equations are used to obtain a new self-similar equation. The latter equation, subject to appropriate boundary conditions and volume discharge, describes the pressure distribution and velocity field of a plane free jet.     

Two-level Galerkin–Lagrange multipliers method for the stationary Navier–Stokes equations

In this paper, we combine the Galerkin–Lagrange multiplier (GLM) method with the two-level method to solve the stationary Navier–Stokes equations in order to avoid the time-consuming process and the construction of zero-divergence elements. Different quadrilateral partitions are used for approximating the velocity and the pressure. Then some error estimates are obtained and some numerical results of the GLM method and the two-level GLM method are given. The results show that the two-level method based on the GLM method is more efficient than the GLM method under the convergence rate of same order.     

Sufficient conditions for the regularity of the solutions of the Navier–Stokes equations

In this paper we find sufficient conditions, involving only the pressure, that ensure the regularity of weak solutions to the Navier–Stokes equations. Conditions involving only the pressure were previously obtained in [7,4]. Following a remark in this last reference we improve, in particular, Kaniel's result [7]. Our condition can be seen at the light of the heuristic idea that the pressure behaves similarly to the modulus squared of the velocity. Copyright © 1999 John Wiley & Sons, Ltd.     

A two-grid method based on Newton iteration for the Navier–Stokes equations

In this paper, we consider a two-grid method for resolving the nonlinearity in finite element approximations of the equilibrium Navier–Stokes equations. We prove the convergence rate of the approximation obtained by this method. The two-grid method involves solving one small, nonlinear coarse mesh system and two linear problems on the fine mesh which have the same stiffness matrix with only different right-hand side. The algorithm we study produces an approximate solution with the optimal asymptotic in h and accuracy for any Reynolds number. Numerical example is given to show the convergence of the method.     

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.     

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.     

Parabolic Dirac operators and the Navier–Stokes equations over time‐varying domains

We consider parabolic Dirac operators which do not involve fractional derivatives and use them to show the solvability of the in‐stationary Navier–Stokes equations over time‐varying domains. Copyright © 2005 John Wiley & Sons, Ltd.     

A remark on the decay of solutions to the 3‐D Navier–Stokes equations

In this paper we derive a decay rate of the L²‐norm of the solution to the 3‐D Navier–Stokes equations. Although the result which is proved by Fourier splitting method is well known, our method is new, concise and direct. Moreover, it turns out that the new method established here has a wide application on other equations. Copyright © 2007 John Wiley & Sons, Ltd.     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

A superconvergent nonconforming mixed finite element method for the Navier–Stokes equations

The superconvergence for a nonconforming mixed finite element approximation of the Navier–Stokes equations is analyzed in this article. The velocity field is approximated by the constrained nonconforming rotated Q₁ (CNRQ₁) element, and the pressure is approximated by the piecewise constant functions. Under some regularity assumptions, the superconvergence estimates for both the velocity in broken H¹‐norm and the pressure in L²‐norm are obtained. Some numerical examples are presented to demonstrate our theoretical results. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 646–660, 2016     

Regularity criteria for the Navier–Stokes equations involving the ratio pressure‐gradient of velocity

A number of bounds upon the pressure are known to guarantee regularity of the solutions of the Navier–Stokes equations. Since the pressure is the potential whose source is the product of the velocity and its gradient, it is worth to consider bounds depending on the quotient of the pressure and some quantity measuring the size of this source. Estimates involving the ratio pressure–velocity are known. Our result includes the velocity gradient: if the ratio remains bounded for some r<1, so does the velocity and therefore it retains its regularity. Copyright © 2009 John Wiley & Sons, Ltd.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Classical solutions to Navier–Stokes equations for nonhomogeneous incompressible fluids with non-negative densities

We study the Navier–Stokes equations for nonhomogeneous incompressible fluids in a bounded domain Ω of R³. We first prove the existence and uniqueness of local classical solutions to the initial boundary value problem of linear Stokes equations and then we obtain the existence and uniqueness of local classical solutions to the Navier–Stokes equations with vacuum under the assumption that the data satisfies a natural compatibility condition.     

Navier–Stokes equations with regularity in one directional derivative of the pressure

This paper concerns the 3D Navier‐Stokes equations and prove an almost Serrin‐type regularity criterion in terms of one directional derivative of the pressure. Copyright © 2014 John Wiley & Sons, Ltd.     

Navier–Stokes equations with degenerate viscosity,vacuum and gravitational force

The global existence of weak solutions to the compressible Navier–Stokes equations with vacuum attracts many research interests nowadays. For the isentropic gas, the viscosity coefficient depends on density function from physical point of view. When the density function connects to vacuum continuously, the vacuum degeneracy gives some analytic difficulties in proving global existence. In this paper, we consider this case with gravitational force and fixed boundary condition. By giving a series of a priori estimates on the solution coping with the degeneracy of vacuum, gravitational force and boundary effect, we give global existence and uniqueness results similar to the case without force and boundary. Copyright © 2006 John Wiley & Sons, Ltd.