Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers

The present paper seeks to continue the analysis in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on tangential boundary stabilization of Navier–Stokes equations, d=2,3$d=2,3$, as deduced from well-posedness and stability properties of the corresponding linearized equations. It intends to complement [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] on two levels: (i) by casting the Riccati-based results of Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] for d=2,3$d=2,3$ in an abstract setting, thus extracting the key relevant features, so that the resulting framework may be applicable also to other stabilizing boundary feedback operators, as well as to other parabolic-like equations of fluid dynamics; (ii) by including, in the case d=2$d=2$ this time, also the low-level gain counterpart of the results in Barbu et al. [Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear] with both Riccati-based and spectral-based (tangential) feedback controllers. This way, new local boundary stabilization results of Navier–Stokes equations are obtained over [V. Barbu, I. Lasiecka, R. Triggiani, Tangential boundary stabilization of Navier–Stokes equations, Memoir AMS, to appear].     

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.     

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.     

Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

The Navier–Stokes limit of the Boltzmann equation for bounded collision kernels

The present work establishes a Navier–Stokes limit for the Boltzmann equation considered over the infinite spatial domain R ³. Appropriately scaled families of DiPerna-Lions renormalized solutions are shown to have fluctuations whose limit points (in the w-L ¹ topology) are governed by Leray solutions of the limiting Navier–Stokes equations. This completes the arguments in Bardos-Golse-Levermore [Commun. Pure Appl. Math. 46(5), 667–753 (1993)] for the steady case, and in Lions-Masmoudi [Arch. Ration. Mech. Anal. 158(3), 173–193 (2001)] for the time-dependent case.Mathematics Subject Classification (2000) 35Q35, 35Q30, 82C40     

Navier–Stokes equations with nonhomogeneous boundary conditions in a convex bi-dimensional domain

We prove the existence of a weak solution to Navier–Stokes equations describing the isentropic flow of a gas in a convex and bounded region, ΩR², with nonhomogeneous Dirichlet boundary conditions on ∂Ω. These results are also extended to flow domain surrounding an obstacle.     

Numerical comparison of nonlinear subgridscale models via adaptive mesh refinement

In this paper, we provide a method of evaluating the efficacy of nonlinear subgridscale models for use in the large eddy simulation of incompressible viscous flow problems. We compare subgridscale “artificial” viscosity models using a posteriori error estimation and adaptive mesh refinement. Specifically, we compare α-Laplacian based subgridscale models and discuss the benefits and limitations of different values of α for some standard benchmark problems for the Navier–Stokes equations.     

Steady flows of Jeffrey–Hamel type from the half-plane into an infinite channel. 1. Linearization on an antisymmetric solution

In this paper our objective is to provide physically reasonable solutions for the stationary Navier–Stokes equations in a two-dimensional domain with two outlets to infinity, a semi-strip Π₋ and a half-plane K. The same problem in an aperture domain, i.e. in a domain with two half-plane outlets to infinity, has been studied but only under symmetry restrictions on the data. Here, we assume that the main asymptotic term of the solution takes an antisymmetric form in K and apply the technique of weighted spaces with detached asymptotics, i.e. we use spaces where the functions have prescribed asymptotic forms in the outlets.After first showing that the corresponding Stokes problem admits a unique solution if and only if certain compatibility conditions are satisfied, we write the Navier–Stokes equations as a perturbation of the Stokes problem and the crucial compatibility condition as an algebraic equation by which the flux becomes determined. Assuming that the coefficient of the main (antisymmetric) asymptotic term of the solution in K does not vanish and that the data are sufficiently small, we use a contraction principle to solve the Navier–Stokes system coupled with the algebraic equation.Finally, we discuss the ill-posedness of the Navier–Stokes problem with prescribed flux.     

Smooth Solution of the Compressible Navier–Stokes Equations in an Unbounded Domain with Inflow Boundary Condition

The barotropic compressible Navier–Stokes equations in an unbounded domain are studied. We prove the unique existence of the solution (u, p) of the system (1.1) in the Sobolev spaceH^{k + 3} × H^{k + 2}provided that the derivatives of the data of the problem are sufficiently small, wherek ≥ 0 is any integer. The proof follows from an analysis of the linearized problem, the solvability of the continuity equation, and the Schauder fixed point theory. Similar smoothness results are obtained for a linearized form of (1.1).     

Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains

We show that the L^p spatial–temporal decay rates of solutions of incompressible flow in an 2D exterior domain. When a domain has a boundary, pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary. To overcome the difficulty, we adopt the ideas in He, Xin [C. He, Z. Xin, Weighted estimates for nonstationary Navier–Stokes equations in exterior domain, Methods Appl. Anal. 7 (3) (2000) 443–458], and our previous results [H.-O. Bae, B.J. Jin, Asymptotic behavior of Stokes solutions in 2D exterior domains, J. Math. Fluid Mech., in press; H.-O. Bae, B.J. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, submitted for publication]. For the spatial decay rate estimate, we first extend temporal decay rate result of the Navier–Stokes solutions for general L^p space when the initial velocity is in , 1<rq<∞ (1<r<q=∞).     

Existence and Regularity of the Pressure for the Stochastic Navier–Stokes Equations

We prove, on one hand, that for a convenient body force with values in the distribution space (H ^-1(D)) ^d , where D is the geometric domain of the fluid, there exist a velocity u and a pressure p solution of the stochastic Navier–Stokes equation in dimension 2, 3 or 4. On the other hand, we prove that, for a body force with values in the dual space V of the divergence free subspace V of (H ¹ ₀(D)) ^d , in general it is not possible to solve the stochastic Navier–Stokes equations. More precisely, although such body forces have been considered, there is no topological space in which Navier–Stokes equations could be meaningful for them.     

Coupled parabolic–hyperbolic Stokes–Lamé PDE system: limit behaviour of the resolvent operator on the imaginary axis

This article is focused on an established, genuinely physical fluid-structure interaction model, whereby the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic–hyperbolic system of two partial differential equations in three dimensions with non-standard coupling at the boundary interface: the (dynamic) Stokes system (parabolic, modelling the fluid) and the Lamé system (hyperbolic, modelling the structure). This system generates a contraction semigroup on the natural energy space [G. Avalos and R. Triggiani, The coupled PDE system arising in fluid/structure interaction, Part I: explicit semigroup generator and its spectral properties, Fluids and Waves, Amer. Math. Soc. Contemp. Math. 440 (2007), pp. 15–59] (canonical model) and [G. Avalos and R. Triggiani, Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction, Discr. Contin. Dyn. Sys. Series S, 2(3) (2009), pp. 417–447]. The boundary interface may or may not include a ‘damping’ (or dissipative) term. If damping is active on the entire interface, then uniform (exponential) stabilization is ensured, regardless of the geometry of the structure [G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22(4) 2008, pp. 817–835, special issue, invited paper] (canonical model) and [G. Avalos and R. Triggiani, Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system, J. Evol. Eqns 9(2009), pp. 341–370]. This article emphasizes the case of, at most, partial damping. At any rate, the main result is a precise uniform-operator limit behaviour of the resolvent operator of the semigroup generator on the imaginary axis of interest in itself, which holds true with or without damping. It, in turn, then implies a fortiori strong stability results: most notably, on the whole state space, under at least partial damping at the interface; and, in the absence of damping, on the whole state space, after factoring out an explicit one-dimensional null eigenspace, at least for a large class of geometries of the structure: these are characterized by a uniqueness property of a special over-determined elliptic problem.     

Local exact controllability of the Navier–Stokes system

In this paper we deal with the local exact controllability of the Navier–Stokes system with distributed controls supported in small sets. In a first step, we present a new Carleman inequality for the linearized Navier–Stokes system, which leads to null controllability at any time T>0. Then, we deduce a local result concerning the exact controllability to the trajectories of the Navier–Stokes system.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction

A 2-d or 3-d fluid-structure interaction model in its linear form is considered, for which semigroup well-posedness (with explicit generator) was recently established in [G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part I: Explicit semigroup generator and its spectral properties, in: Fluids and Waves, in: Contemp. Math., vol. 440, Amer. Math. Soc., 2007, pp. 15-55; G. Avalos, R. Triggiani, The coupled PDE-system arising in fluid-structure interaction. Part II: Uniform stabilization with boundary dissipation at the interface, Discrete Contin. Dyn. Syst., in press]. This is a system which couples at the interface the linear version of the Navier-Stokes equations with the equations of linear elasticity (wave-like). In this paper, we establish a backward uniqueness theorem for such a parabolic-hyperbolic coupled PDE system. If {eAt}_t?0 is the (contraction) s.c. semigroup describing its evolution on the finite energy space H, then eATy₀=0 for some T>0 and y₀∈H, implies y₀=0. This property has implications in establishing unique continuation and controllability properties, as in the case of thermoelastic equations [M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, in: Marcel Dekker Lect. Notes Pure Appl. Math., vol. 216, February 2001, pp. 109-230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol, J.P. Zolesio (Eds). (Preliminary version is in invited paper in: A.V. Balakrishnan (Ed.), Semigroup of Operators and Applications, Birkhäuser, 2000, pp. 335-351.); M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000) 452-478; M. Eller, I. Lasiecka, R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls, Discrete Contin. Dyn. Syst. 7 (2) (2001) 283-301].     

Analysis of a two-dimensional grade-two fluid model with a tangential boundary condition

This article studies the solutions in H¹ of a two-dimensional grade-two fluid model with a non-homogeneous Dirichlet tangential boundary condition, on a Lipschitz-continuous domain. Existence is proven by splitting the problem into a generalized Stokes problem and a transport equation, without restricting the size of the data and the constant parameters of the fluid. A substantial part of the article is devoted to a sharp analysis of this transport equation, under weak regularity assumptions. By means of this analysis, it is established that each solution of the grade-two fluid model satisfies energy equalities and converges strongly to a solution of the Navier–Stokes equations when the normal stress modulus α tends to zero. When the domain is a polygon, it is shown that the regularity of the solution is related to that of a Stokes problem. Uniqueness is established in a convex polygon, with adequate restrictions on the size of the data and parameters.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

Vertex Set Partitions Preserving Conservativeness

Let G be an undirected graph and ={X₁, …, X_n} be a partition of V(G). Denote by G/ the graph which has vertex set {X₁, …, X_n}, edge set E, and is obtained from G by identifying vertices in each class X_i of the partition . Given a conservative graph (G, w), we study vertex set partitions preserving conservativeness, i.e., those for which (G/ , w) is also a conservative graph. We characterize the conservative graphs (G/ , w), where is a terminal partition of V(G) (a partition preserving conservativeness which is not a refinement of any other partition of this kind). We prove that many conservative graphs admit terminal partitions with some additional properties. The results obtained are then used in new unified short proofs for a co-NP characterization of Seymour graphs by A. A. Ageev, A. V. Kostochka, and Z. Szigeti (1997, J. Graph Theory34, 357–364), a theorem of E. Korach and M. Penn (1992, Math. Programming55, 183–191), a theorem of E. Korach (1994, J. Combin. Theory Ser. B62, 1–10), and a theorem of A. V. Kostochka (1994, in “Discrete Analysis and Operations Research. Mathematics and its Applications (A. D. Korshunov, Ed.), Vol. 355, pp. 109–123, Kluwer Academic, Dordrecht).     

On Backward Uniqueness for Parabolic Equations

We prove a backward uniqueness result for the heat operator with variable lower order terms, which implies the full regularity of L _3,-solutions of the three-dimensional Navier–Stokes equations. Bibliography: 12 titles.     

Existence and Regularity of a Class of Weak Solutions to the Navier–Stokes Equations

We construct a class of weak solutions to the Navier–Stokes equations, which have second order spatial derivatives and one order time derivatives, ofppower summability for 1 < p ≤ 5/4. Meanwhile, we show thatu L^s(0, T; W^{2, r}(Ω)) with 1/s + 3/2r = 2 for 1 < r ≤ 5/4.rcan be relaxed not to exceed 3/2 if we consider only in the interior of Ω. In the end, we extend the classical regularity theorem. Our results show thatuis a regular solution if u L^s(0, T; L^r(Ω)) with 1/s + 3/2r = 1 for Ω satisfying (1.3), with 1/s + 1/r = 5/6 for arbitrary domain inR³and 1 < s ≤ 2. For Ω = Rⁿwithn ≥ 3, this result was previously obtained byH. Beirão da Veiga (Chinese Ann. Math. Ser. B16, 1995, 407–412).