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Volume 53, Issue 11, November 2012
 SPECIAL ISSUE: INCOMPRESSIBLE FLUIDS, TURBULENCE AND MIXING


Introduction to Special Issue: Incompressible Fluids, Turbulence and Mixing
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Logarithmically regularized inviscid models in borderline sobolev spaces
View Description Hide DescriptionSeveral inviscid models in hydrodynamics and geophysics such as the incompressible Euler vorticityequations, the surface quasigeostrophic equation, and the Boussinesq equations are not known to have even local wellposedness in the corresponding borderline Sobolev spaces. Here H ^{ s } is referred to as a borderline Sobolev space if the L ^{∞}norm of the gradient of the velocity is not bounded by the H ^{ s }norm of the solution but by the norm for any . This paper establishes the local wellposedness of the logarithmically regularized counterparts of these inviscid models in the borderline Sobolev spaces.

On a singular incompressible porous media equation
View Description Hide DescriptionThis paper considers a family of active scalar equations with transport velocities which are more singular by a derivative of order β than the active scalar. We prove that the equations with 0 < β ⩽ 2 are Lipschitz illposed for regular initial data. On the contrary, when 0 < β < 1 we show local wellposedness for patchtype weak solutions.

Longtime behavior of a twolayer model of baroclinic quasigeostrophic turbulence
View Description Hide DescriptionWe study a viscous twolayer quasigeostrophic betaplane model that is forced by imposition of a spatially uniform vertical shear in the eastward (zonal) component of the layer flows, or equivalently a spatially uniform northsouth temperature gradient. We prove that the model is linearly unstable, but that nonlinear solutions are bounded in time by a bound which is independent of the initial data and is determined only by the physical parameters of the model. We further prove, using arguments first presented in the study of the Kuramoto–Sivashinsky equation, the existence of an absorbing ball in appropriate function spaces, and in fact the existence of a compact finitedimensional attractor, and provide upper bounds for the fractal and Hausdorff dimensions of the attractor. Finally, we show the existence of an inertial manifold for the dynamical system generated by the model'ssolution operator. Our results provide rigorous justification for observations made by Panetta based on longtime numerical integrations of the model equations.

Liouville theorems in unbounded domains for the timedependent stokes system
View Description Hide DescriptionIn this paper, we characterize bounded ancient solutions to the timedependent Stokes system with zero boundary value in various domains, including the halfspace.

Coercivity and stability results for an extended NavierStokes system
View Description Hide DescriptionIn this paper, we study a system of equations that is known to extendNavierStokes dynamics in a wellposed manner to velocity fields that are not necessarily divergencefree. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of both controlling the nonlinear terms, and being useful at the timediscrete level. We address questions of global existence and stability in bounded domains with noslip boundary conditions. Through use of new H ^{1}coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a timediscrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.

Wellposedness on large time for a modified full dispersion system of surface waves
View Description Hide DescriptionBy a nonlinear change of variables from the original one, we derive a “small steepness full dispersion” system for surface waterwaves which is consistent with the waterwavesystem. This system is symmetrizable and we prove that the Cauchy problem is wellposed on large time of order 1/ε where ε is the steepness coefficient, implying (together with the results of D. Lannes) its full rigorous justification as an asymptotic model to the full Euler equations with free surface.

The FeffermanStein decomposition for the ConstantinLaxMajda equation: Regularity criteria for inviscid fluid dynamics revisited
View Description Hide DescriptionThe celebrated BealeKatoMajda (BKM) criterion for the 3D Euler equations has been updated by Kozono and Taniuchi by replacing the supremum with the bounded mean oscillation norm. We consider this generalized criterion in an attempt to understand it more intuitively by giving an alternative explanation. For simplicity, we first treat the ConstantinLaxMajda (CLM) equation for the vorticity ω in onedimension and identify a mechanism underlying the update of such an estimate. We consider a FeffermanStein (FS) decomposition for the initial vorticity ω = ω_{0} + H[ω_{1}] and how it propagates under the dynamics of the CLM equation. In particular, we obtain a set of dynamical equations for it, which reads in its simplest case and . The equation for the second component ω_{1}, responsible for a possible logarithmic blowup, is linear and homogeneous; hence it remains zero if it is so initially until a stronger blowup takes place. This rules out a logarithmic blowup on its own and underlies the generalized BKM criterion. Numerical results are also presented to illustrate how each component of the FS decomposition evolves in time. Higher dimensional cases are also discussed. Without knowing fully explicit FS decompositions for the 3D Euler equations, we show that the second component of the FS decomposition will not appear if it is zero initially, thereby precluding a logarithmic blowup.

Conditional regularity of solutions of the threedimensional NavierStokes equations and implications for intermittency
View Description Hide DescriptionTwo unusual timeintegral conditional regularity results are presented for the threedimensional NavierStokes equations. The ideas are based on L ^{2m }norms of the vorticity, denoted by Ω_{ m }(t), and particularly on , where α_{ m } = 2m/(4m − 3) for m ⩾ 1. The first result, more appropriate for the unforced case, can be stated simply: if there exists an 1 ⩽ m < ∞ for which the integral condition is satisfied (Z _{ m } = D _{ m + 1}/D _{ m }): , then no singularity can occur on [0, t]. The constant c _{4, m } ↘ 2 for large m. Second, for the forced case, by imposing a critical lower bound on , no singularity can occur in D _{ m }(t) for large initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive over this critical value can be ruled out whereas other types cannot.

Biomixing by chemotaxis and efficiency of biological reactions: The critical reaction case
View Description Hide DescriptionMany phenomena in biology involve both reactions and chemotaxis. These processes can clearly influence each other, and chemotaxis can play an important role in sustaining and speeding up the reaction. In continuation of our work [A. Kiselev and L. Ryzhik, “Biomixing by chemotaxis and enhancement of biological reactions,” Comm. Partial Differential Equations37, 298–318 (2012)]10.1080/03605302.2011.589879, we consider a model with a single density function involving diffusion, advection, chemotaxis, and absorbing reaction. The model is motivated, in particular, by the studies of coral broadcast spawning, where experimental observations of the efficiency of fertilization rates significantly exceed the data obtained from numerical models that do not take chemotaxis (attraction of sperm gametes by a chemical secreted by egg gametes) into account. We consider the case of the weakly coupled quadratic reaction term, which is the most natural from the biological point of view and was left open in Kiselev and Ryzhik [“Biomixing by chemotaxis and enhancement of biological reactions,” Comm. Partial Differential Equations37, 298–318 (2012)]10.1080/03605302.2011.589879. The result is that similarly to Kiselev and Ryzhik [“Biomixing by chemotaxis and enhancement of biological reactions,” Comm. Partial Differential Equations37, 298–318 (2012)]10.1080/03605302.2011.589879, the chemotaxis plays a crucial role in ensuring efficiency of reaction. However, mathematically, the picture is quite different in the quadratic reaction case and is more subtle. The reaction is now complete even in the absence of chemotaxis, but the timescales are very different. Without chemotaxis, the reaction is very slow, especially for the weak reaction coupling. With chemotaxis, the timescale and efficiency of reaction are independent of the coupling parameter.

Stability and clustering of selfsimilar solutions of aggregation equations
View Description Hide DescriptionIn this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρ_{ t } = ∇ · (ρ∇K * ρ) in , d ⩾ 2, where K(r) = r ^{γ}/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Selfsimilar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math.70, 2582–2603 (Year: 2010)]10.1137/090774495 that radially symmetric solutions are attracted to a selfsimilar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ > 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that twodimensional (in)stability implies ndimensional (in)stability.

Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows
View Description Hide DescriptionWe consider passive scalar mixing by a prescribed divergencefree velocity vector field in a periodic box and address the following question: Starting from a given initial inhomogeneous distribution of passive tracers, and given a certain energy budget, power budget, or finite palenstrophy budget, what incompressible flow field best mixes the scalar quantity? We focus on the optimal stirring strategy recently proposed by Lin et al. [“Optimal stirring strategies for passive scalar mixing,” J. Fluid Mech.675, 465 (2011)]10.1017/S0022112011000292 that determines the flow field that instantaneously maximizes the depletion of the H ^{−1} mixnorm. In this work, we bridge some of the gap between the best available a priori analysis and simulation results. After recalling some previous analysis, we present an explicit example demonstrating finitetime perfect mixing with a finite energy constraint on the stirring flow. On the other hand, using a recent result by Wirosoetisno et al. [“Long time stability of a classical efficient scheme for two dimensional NavierStokes equations,” SIAM J. Numer. Anal.50(1), 126–150 (2012)]10.1137/110834901 we establish that the H ^{−1} mixnorm decays at most exponentially in time if the twodimensional incompressible flow is constrained to have constant palenstrophy. Finitetime perfect mixing is thus ruled out when too much cost is incurred by small scale structures in the stirring. Direct numerical simulations in two dimensions suggest the impossibility of finitetime perfect mixing for flows with fixed power constraint and we conjecture an exponential lower bound on the H ^{−1} mixnorm in this case. We also discuss some related problems from other areas of analysis that are similarly suggestive of an exponential lower bound for the H ^{−1} mixnorm.

An initial and boundaryvalue problem for the ZakharovKuznestov equation in a bounded domain
View Description Hide DescriptionMotivated by the study of boundary control problems for the ZakharovKuznetsov equation, we study in this article the initial and boundary value problem for the ZK (short for ZakharovKuznetsov) equation posed in a limited domain Ω = (0, 1)_{ x } × (−π/2, π/2)^{ d }, d = 1, 2. This article is related to Saut and Temam [“An initial boundaryvalue problem for the ZakharovKuznetsov equation,” Adv. Differ. Equ.15(11–12), 1001–1031 (2010)] in which the authors studied the same problem in the band d = 1, 2, but this article is not a straightforward adaptation of Saut and Temam [“An initial boundaryvalue problem for the ZakharovKuznetsov equation,” Adv. Differ. Equ.15(11–12), 1001–1031 (2010)]; indeed many new issues arise, in particular, for the function spaces, due to the loss of the Fourier transform in the tangential directions (orthogonal to 0x). In this article, after studying a number of suitable function spaces, we show the existence and uniqueness of solutions for the linearized equation using the linear semigroup theory. We then continue with the nonlinear equation with the homogeneous boundary conditions. The case of the full nonlinear equation with nonhomogeneous boundary conditions especially needed for the control problems will be studied elsewhere.

Vortex stretching and criticality for the threedimensional NavierStokes equations
View Description Hide DescriptionA mathematical evidence—in a statistically significant sense—of a geometric scenario leading to criticality of the NavierStokes problem is presented.

Multiscale turbulence models based on convected fluid microstructure
View Description Hide DescriptionThe EulerPoincaré approach to complex fluids is used to derive multiscale equations for computationally modeling Euler flows as a basis for modelingturbulence. The model is based on a kinematic sweeping ansatz (KSA) which assumes that the mean fluid flow serves as a Lagrangian frame of motion for the fluctuationdynamics. Thus, we regard the motion of a fluid parcel on the computationally resolvable length scales as a moving Lagrange coordinate for the fluctuating (zeromean) motion of fluid parcels at the unresolved scales. Even in the simplest twoscale version on which we concentrate here, the contributions of the fluctuating motion under the KSA to the mean motion yields a system of equations that extends known results and appears to be suitable for modeling nonlinear backscatter (energy transfer from smaller to larger scales) in turbulence using multiscale methods.

Stability and the continuum limit of the spinpolarized ThomasFermiDiracvon Weizsäcker model
View Description Hide DescriptionThe continuum limit of the spinpolarized ThomasFermiDiracvon Weizsäcker model in an external magnetic field is studied. An extension of the classical CauchyBorn rule for crystal lattices is established for the electronic structure under sharp stability conditions on charge density and spin density waves. A LandauLifshitz type of micromagnetic energy functional is derived.

External noise control in inherently stochastic biological systems
View Description Hide DescriptionBiological systems are often subject to external noise from signal stimuli and environmental perturbations, as well as noises in the intracellular signal transduction pathway. Can different stochastic fluctuations interact to give rise to new emerging behaviors? How can a system reduce noise effects while still being capable of detecting changes in the input signal? Here, we study analytically and computationally the role of nonlinear feedback systems in controlling external noise with the presence of large internal noise. In addition to noise attenuation, we analyze derivatives of Fano factor to study systems' capability of differentiating signal inputs. We find effects of internal noise and external noise may be separated in one slow positive feedback loop system; in particular, the slow loop can decrease external noise and increase robustness of signaling with respect to fluctuations in rate constants, while maintaining the signal output specific to the input. For two feedback loops, we demonstrate that the influence of external noise mainly depends on how the fast loop responds to fluctuations in the input and the slow loop plays a limited role in determining the signal precision. Furthermore, in a dual loop system of one positive feedback and one negative feedback, a slower positive feedback always leads to better noise attenuation; in contrast, a slower negative feedback may not be more beneficial. Our results reveal interesting stochastic effects for systems containing both extrinsic and intrinsic noises, suggesting novel noise filtering strategies in inherently stochastic systems.

A new boundary condition for the threedimensional NavierStokes equation and the vanishing viscosity limit
View Description Hide DescriptionIn this paper, we propose a new vorticity boundary condition for the threedimensional incompressible NavierStokes equation for a general smooth domain in R ^{3}. This boundary condition is motivated by the generalized Navierslip boundary condition involving the vorticity. It is shown first that such an initial boundary value problem is wellposed at least local in time. Furthermore, more importantly, we obtain some estimates on rate of convergence in C([0, T], H ^{1}(Ω)) and C([0, T], H ^{2}(Ω)) of the solutions to the corresponding solutions of the ideal Euler equations with the standard slip boundary condition.

Lower bounds on blow up solutions of the threedimensional Navier–Stokes equations in homogeneous Sobolev spaces
View Description Hide DescriptionSuppose that u(t) is a solution of the threedimensional Navier–Stokes equations, either on the whole space or with periodic boundary conditions, that has a singularity at time T. In this paper we show that the norm of u(T − t) in the homogeneous Sobolev space must be bounded below by c _{ s } t ^{−(2s−1)/4} for 1/2 < s < 5/2 (s ≠ 3/2), where c _{ s } is an absolute constant depending only on s; and by for s > 5/2. (The result for 1/2 < s < 3/2 follows from wellknown lower bounds on blowup in L ^{ p } spaces.) We show in particular that the local existence time in depends only on the norm for 1/2 < s < 5/2, s ≠ 3/2.

Spreading speeds for onedimensional monostable reactiondiffusion equations
View Description Hide DescriptionWe establish in this article spreading properties for the solutions of equations of the type ∂_{ t } u − a(x)∂_{ xx } u − q(x)∂_{ x } u = f(x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable Kolmogorov, Petrovsky, and Piskunov type between two steady states 0 and 1 and the initial datum is compactly supported. Using homogenization techniques, we construct two speeds such that for all and for all . These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where ).
