Global solution of 3D incompressible magnetohydrodynamics equations with finite energy

We construct a family of finite energy classical solutions to the 3D MHD system with both Laplacian dissipation and magnetic diffusion. We choose the steady state Beltrami flows as the initial data and use a cut-off technique to obtain the global regularity for all time t.     

Lifespan Theorem for simple constrained surface diffusion flows

We consider closed immersed hypersurfaces in R³ and R⁴ evolving by a special class of constrained surface diffusion flows. This class of constrained flows includes the classical surface diffusion flow. In this paper we present a Lifespan Theorem for these flows, which gives a positive lower bound on the time for which a smooth solution exists, and a small upper bound on the total curvature during this time. The hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature. This turns out to be a deep property of the initial manifold, as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in L² for M² immersed in R³ and additionally on the concentration in L³ for M³ immersed in R⁴. This is stronger than a previous result on a different class of constrained surface diffusion flows, as here we obtain an improved lower bound on maximal time, a better estimate during this period, and eliminate any assumption on the area of the evolving hypersurface.     

On weighted-norm estimates for nonstationary incompressible Navier-Stokes flows in a 3D exterior domain

We study the time-decay of weighted norms of weak and strong solutions to the Navier-Stokes equations in a 3D exterior domain. Moment estimates for weak solutions and weighted Lq-estimates for strong solutions are deduced, both of which seem to be optimal. The relation is discussed between the space-time decay and the vanishing of the total net force exerted by the fluid to the body. A class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish.     

Large time behavior for the incompressible Navier–Stokes flows in 2D exterior domains

The incompressible Navier–Stokes flows in a 2D exterior domain are considered, for which the associated total net force to the boundary may not vanish. The decay properties are shown for the first and second derivatives of the Navier–Stokes flows in L ¹ and weighted spaces, respectively, which improve Theorem 1.2 in Bae and Jin (J Funct Anal 240:508–529, 2006).     

Compactness of weak solutions to the three-dimensional compressible magnetohydrodynamic equations

The compactness of weak solutions to the magnetohydrodynamic equations for the viscous, compressible, heat conducting fluids is considered in both the three-dimensional space R³ and the three-dimensional periodic domains. The viscosities, the heat conductivity as well as the magnetic coefficient are allowed to depend on the density, and may vanish on the vacuum. This paper provides a different idea from [X. Hu, D. Wang, Global solutions to the three-dimensional full compressible magnetohydrodynamic flows, Comm. Math. Phys. (2008), in press] to show the compactness of solutions of viscous, compressible, heat conducting magnetohydrodynamic flows, derives a new entropy identity, and shows that the limit of a sequence of weak solutions is still a weak solution to the compressible magnetohydrodynamic equations.     

Modifying a branched surface to carry a foliation

We consider C¹ nonsingular flows on a closed 3-manifold under which there is no transverse disk that flows continuously back into its own interior. We provide an algorithm for modifying any branched surface transverse to such a flow ? that terminates in a branched surface carrying a foliation F precisely when F is transverse to ?. As a corollary, we find branched surfaces that do not carry foliations but that lift to ones that do.     

On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces

This paper studies the regularity criterion of weak solutions for three-dimensional (3D) micropolar fluid flows. When the velocity field satisfies for −1<r<1, then the weak solution (u,w) is regular on (0,T]. The methods are mainly based on the Fourier localization technique and Bony’s para-product decomposition.     

Construction of a high order fluid-structure interaction solver

Accuracy is critical if we are to trust simulation predictions. In settings such as fluid-structure interaction, it is all the more important to obtain reliable results to understand, for example, the impact of pathologies on blood flows in the cardiovascular system. In this paper, we propose a computational strategy for simulating fluid structure interaction using high order methods in space and time.First, we present the mathematical and computational core framework, Life, underlying our multi-physics solvers. Life is a versatile library allowing for 1D, 2D and 3D partial differential solves using h/p type Galerkin methods. Then, we briefly describe the handling of high order geometry and the structure solver. Next we outline the high-order space-time approximation of the incompressible Navier-Stokes equations and comment on the algebraic system and the preconditioning strategy. Finally, we present the high-order Arbitrary Lagrangian Eulerian (ALE) framework in which we solve the fluid-structure interaction problem as well as some initial results.     

Assessment of subgrid-scale models for the incompressible Navier-Stokes equations

In this paper, we assess two kinds of subgrid finite element methods for the two-dimensional (2D) incompressible Naver-Stokes equations (NSEs). These methods introduce subgrid-scale (SGS) eddy viscosity terms which do not act on the large flow structures. The eddy viscous terms consist of the fluid flow fluctuation strain rate stress tensors. The fluctuation tensor can be calculated by a elliptic projection or a simple L² projection (projective filter) in finite element spaces. The finite element pair P₂/P₁ is adopted to numerically implement analysis and computation. We give a complete error analysis based on the assumptions of some regularity conditions. On the part of numerical tests, the numerical computations for the stationary flows show that the numerical results agree with some benchmark solutions and theoretical analysis very well. Furthermore, the given SGS models are applied to the non-stationary fluid flows.     

A fuzzy approach to R&D project portfolio selection

A major advance in the development of project selection tools came with the application of options reasoning in the field of Research and Development (R&D). The options approach to project evaluation seeks to correct the deficiencies of traditional methods of valuation through the recognition that managerial flexibility can bring significant value to projects. Our main concern is how to deal with non-statistical imprecision we encounter when judging or estimating future cash flows. In this paper, we develop a methodology for valuing options on R&D projects, when future cash flows are estimated by trapezoidal fuzzy numbers. In particular, we present a fuzzy mixed integer programming model for the R&D optimal portfolio selection problem, and discuss how our methodology can be used to build decision support tools for optimal R&D project selection in a corporate environment.     

Self-similar flows

Large classes of self-similar (isospectral) flows can be viewed as continuous analogues of certain matrix eigenvalue algorithms. In particular there exist families of flows associated with the QR, LR, and Cholesky eigenvalue algorithms. This paper uses Lie theory to develop a general theory of self-similar flows which includes the QR, LR, and Cholesky flows as special cases. Also included are new families of flows associated with the SR and HR eigenvalue algorithms. The basic theory produces analogues of unshifted, single-step eigenvalue algorithms, but it is also shown how the theory can be extended to include flows which are continuous analogues of shifted and multiple-step eigenvalue algorithms.     

Distributional Enstrophy Dissipation Via the Collapse of Three Point Vortices

Dissipation of enstrophy in 2D incompressible flows in the zero viscous limit is considered to play a significant role in the emergence of the inertial range corresponding to the forward enstrophy cascade in the energy spectrum of 2D turbulent flows. However, since smooth solutions of the 2D incompressible Euler equations conserve the enstrophy, we need to consider non-smooth inviscid and incompressible flows so that the enstrophy dissipates. Moreover, it is physically uncertain what kind of a flow evolution gives rise to such an anomalous enstrophy dissipation. In this paper, in order to acquire an insight about the singular phenomenon mathematically as well as physically, we consider a dispersive regularization of the 2D Euler equations, known as the Euler-\(\alpha \) equations, for the initial vorticity distributions whose support consists of three points, i.e., three \(\alpha \)-point vortices, and take the \(\alpha \rightarrow 0\) limit of its global solutions. We prove with mathematical rigor that, under a certain condition on their vortex strengths, the limit solution becomes a self-similar evolution collapsing to a point followed by the expansion from the collapse point to infinity for a wide range of initial configurations of point vortices. We also find that the enstrophy always dissipates in the sense of distributions at the collapse time. This indicates that the triple collapse is a mechanism for the anomalous enstrophy dissipation in non-smooth inviscid and incompressible flows. Furthermore, it is an interesting example elucidating the emergence of the irreversibility of time in a Hamiltonian dynamical system.     

Classification of gradient-like flows on dimensions two and three

We consider here gradient-like flows of classC ^r ( ^r ≥1) on a closed manifoldM of classC ^r+1 and dimension two or three. We study the classification of these flows by the relation of topological equivalence. In this sense, the flows which are more relevant are the polar flows (only one source and only one sink).     

An adaptive grid technique for the vertical structure of shallow water models

The efficient resolution of the boundary layers occurring in geophysical flows has motivated the search for criteria to optimize the vertical nodal placement in three-dimensional (3D) shallow water models. This paper describes the implementation and testing of an adaptive grid technique for the internal mode of shallow water models. The technique uses an r-method in which the nodes are moved vertically based on the velocity gradients between consecutive nodes. One-dimensional (1D) tests show that the method behaves well in tidal- and wind-driven flows, both in well-mixed and stratified conditions. Average accuracy improvements of 50% were obtained relative to uniform grids, with a 15% CPU time increase. The adaptive technique accounts accurately for the space and time variability of the flow, thus being attractive for any type of problem. Furthermore, the technique does not require an a priori knowledge of the flow conditions, thus simplifying greatly the modeling procedure.     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Hyperbolic sets for semilinear parabolic equations

In this note we considerC ^r semiflows on Banach spaces, roughly speakingC ^r flows defined only for positive values of time. Such semiflows arise as the “general solution” of a large class of partial differential equations that includes the Navier-Stokes equation. Our main result (Proposition B) is that under certain assumptions on the P.D.E. (satisfield by the Navier-Stokes equation) a hyperbolic set for the corresponding semiflow (hyperbolicity is defined following closely the finite dimensional case) is always ε-equivalent to a hyperbolic set for an ordinary differential equation that can be easily deduced from the P.D.E. As an example we consider the P.D.E. (0) $$\frac{{\partial u}}{{\partial t}} = - \Delta u + \varepsilon F(x,u,u')$$ where u:M → ? ^k andM is a closed smooth Riemannian manifold. Applying normal hyperbolicity techniques the phase portrait of (0) can be analyzed proving that every example of hyperbolic set for O.D.E. can appear as a hyperbolic set for the semiflow generated by (0).     

The Bernoulli Integral for a Certain Class of Non‐Stationary Viscous Vortical Flows of Incompressible Fluid

It has been shown in our paper [1] that there is a wide class of 3D motions of incompressible viscous fluid which can be described by one scalar function dabbed the quasi‐potential. This class of fluid flows is characterized by three‐component velocity field having two‐component vorticity field; both these fields can depend of all three spatial variables and time, in general. Governing equations for the quasi‐potential have been derived and simple illustrative example of 3D flow has been presented. Here, we derive the Bernoulli integral for that class of flows and compare it against the known Bernoulli integrals for the potential flows or 2D stationary vortical flows of inviscid fluid. We show that the Bernoulli integral for this class of fluid motion possesses unusual features: it is valid for the vortical nonstationary motions of a viscous incompressible fluid. We present a new very nontrivial analytical example of 3D flow with two‐component vorticity which hardly can be obtained by any of known methods. In the last section, we suggest a generalization of the developed concept which allows one to describe a certain class of 3D flows with the 3D vorticity.     

Nonexistence of Bonatti-Langevin examples of Anosov flows on closed four manifolds

Bonatti and Langevin constructed an Anosov flow on a closed 3-manifold with a transverse torus intersecting all orbits except one [C. Bonatti, R. Langevin, Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension, Ergodic Theory Dynam. Systems 14 (4) (1994), 633-643]. We shall prove that these flows cannot be constructed on closed 4-manifolds. More precisely, there are no Anosov flows on closed 4-manifolds with a closed, incompressible, transverse submanifold intersecting all orbits except finitely many closed ones. The proof relies on the analysis of the trace of the weak invariant foliations of the flow on the transverse submanifold.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.