Global regularity for the 2D Boussinesq equations with partial viscosity terms

In this paper, we prove the global in time regularity for the 2D Boussinesq system with either the zero diffusivity or the zero viscosity. We also prove that as diffusivity (viscosity) tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion (viscosity) equations. Our result for the zero diffusion system, in particular, solves the Problem no. 3 posed by Moffatt in [R.L. Ricca, (Ed.), Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001, pp. 3-10].     

Non-Uniform Decay of MHD Equations With and Without Magnetic Diffusion

We consider the long time behavior of solutions to the magnetohydrodynamics equations in two and three spatial dimensions. It is shown that in the absence of magnetic diffusion, if strong bounded solutions were to exist their energy cannot present any asymptotic oscillatory behavior, the diffusivity of the velocity is enough to prevent such oscillations. When magnetic diffusion is present and the data is only in L ², it is shown that the solutions decay to zero without a rate, and this nonuniform decay is optimal.     

Homogenization of periodic linear degenerate PDEs

It is well known under the name of ‘periodic homogenization’ that, under a centering condition of the drift, a periodic diffusion process on Rd converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs.     

Diffusion approximation of branching migration processes

We study the behavior of a Galton-Watson process with homogeneous migration component stopped at zero (i.e., the state zero is absorbing). Assuming that the process is initiated at time zero by a large number of particles, we find a diffusion approximation for this process in the case where the average number of offspring per individual is close to one. Supported by the Russian Foundation for Fundamental Research (grant Nos. 96-01-00338 and 96-15-96092) and INTAS-RFBR (grant No. 95-0099). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

流形上随机流的渐近平坦性

本文的目的是讨论流形上由随机微分方程确定的扩散过程的体积零化性质。令X_t(x)是描述流形M上的微分同胚流x→X_t(x)的扩散过程,K是M中具有正有限Hausdorff测度的紧致曲面,我们给出X_t(K)的面积在t→∞时几乎必然趋于零的条件,特别地,随机流X_t(·)的渐近零化定向可求长弧r:[0,1]→M的弧长。     

The Effect of Mean Flows on Enhanced Diffusivity in Transport by Incompressible Periodic Velocity Fields

Avellaneda and one of the authors ([1], [3]) have recently established that an upper bound for the enhanced diffusivity in the large scale, long time advection-diffusion with periodic steady incompressible velocity fields has the form where Pe is the Peclet number and is the reciprocal of the Prandtl number. In this paper, flow fields with maximal and minimal enhanced diffusion are studied. Maximal enhanced diffusion requires that in some directions the enhanced diffusion tensor also has the lower bound . For minimal enhanced diffusion, the effect of the velocity field is to boost the enhanced diffusivity by a negligible amount that is bounded by a fixed constant times the bare diffusivity regardless of Peclet number. Stieltjes measure formulas are used to develop a simple, necessary, and sufficient condition for maximal enhanced diffusion and also to characterize minimal enhanced diffusion. It is established here that constant mean flows can have a dramatic effect on maximal and minimal enhanced diffusion. In particular, for flows in two space dimensions, an explicit criterion is developed that guarantees the surprising fact that mean flows with rational ratios typically generate maximal enhanced diffusion through interaction with an arbitrary steady periodic incompressible flow with zero mean. In contrast, a simple criterion for flows without stagnation points is developed here that guarantees that the effect of mean flows with irrational ratios on advection-diffusion in two dimensions creates minimal enhanced diffusion. The theory for the phenomena mentioned above is elementary yet mathematically rigorous. Examples are emphasized throughout this work including a discussion of enhanced diffusivity for a class of flows recently introduced by Childress and Soward [8]. The theory developed here is also supplemented by a series of numerical experiments that both verify the theoretical predictions and display interesting crossover phenomena at rather large but finite Peclet numbers.     

On dispersion of settling particles from an elevated source in an open-channel flow

Longitudinal dispersion of suspended particles with settling velocity in a turbulent shear flow over a rough-bed surface is investigated numerically when the settling particles are released from an elevated continuous line-source. A combined scheme of central and four-point upwind differences is used to solve the steady turbulent convection–diffusion equation and the alternating direction implicit (ADI) method is adopted for the unsteady equation. It is shown how the mixing of settling particles is influenced by the ‘log-wake law’ velocity and the corresponding eddy diffusivity when the initial distribution of concentration is regarded as a line-source. The concentration profiles for the steady-state conditions agree well with the existing experimental data and some other numerical results when the settling velocity is zero. The behaviours of iso-concentration lines in the vertical plane for different releasing heights are studied in terms of the relative importance of convection, eddy diffusion and settling velocity.     

The diffusive form of Richards’ equation with hysteresis

Coupling hysteretic hydraulic laws with the pressure form of Richards’ equation, a mathematical model of a hysteretic wetting–drying cycle of a soil is settled. The particularity of the model resides in the blowing-up diffusion coefficient characterizing a strongly nonlinear behavior of the porous medium and in certain relationships between the hydraulic functions accounting for a sufficiently realistic hysteretic evolution of the envisaged process. The hysteretic effect of the hydraulic laws can be regained in the hysteretic behavior of the multivalued function defined as an antiderivative of the diffusivity function. We investigate the well-posedness of the model in appropriate functional spaces.     

Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation

A model associated with the formation of sedimentary ocean deltas is presented. This model is a generalized one-dimensional Stefan problem bounded by two moving boundaries, the shoreline and the alluvial-bedrock transition. The sediment transport is a non-linear diffusive process; the diffusivity modeled as a power law of the fluvial slope. Dimensional analysis shows that the first order behavior of the moving boundaries is determined by the dimensionless parameter 0?Rab?1—the ratio of the fluvial slope to bedrock slope at the alluvial-bedrock transition. A similarity form of the governing equations is derived and a solution that tracks the boundaries obtained via the use of a numerical ODE solver; in the cases where the exponent θ in the diffusivity model is zero (linear diffusion) or infinite, closed from solutions are found. For the full range of the diffusivity exponents, 0?θ→∞, the similarity solution shows that when Rab<0.4 there is no distinction in the predicted speeds of the moving boundaries. Further, within the range of physically meaningful values of the diffusivity exponent, i.e., 0?θ∼2, reasonable agreement in predictions extents up to Rab∼0.7. In addition to the similarity solution a fixed grid enthalpy like solution is also proposed; predictions obtained with this solution closely match those obtained with the similarity solution.     

The effect of turbulence on mixing in prototype reaction‐diffusion systems

The effect of turbulence on mixing in prototype reaction‐diffusion systems is analyzed here in the special situation where the turbulence is modeled ideally with two separated scales consisting of a large‐scale mean flow plus a small‐scale spatiotemporal periodic flow. In the limit of fast reaction and slow diffusion, it is rigorously proved that the turbulence does not contribute to the location of the mixing zone in the limit and that this mixing zone location is determined solely by advection of the large‐scale velocity field. This surprising result contrasts strongly with earlier work of the authors that always yields a large‐scale propagation speed enhanced by small‐scale turbulence for propagating fronts. The mathematical reasons for these differences are pointed out. This main theorem rigorously justifies the limit equilibrium approximations utilized in non‐premixed turbulent diffusion flames and condensation‐evaporation modeling in cloud physics in the fast reaction limit. The subtle nature of this result is emphasized by explicit examples presented in the fast reaction and zero‐diffusion limit with a nontrivial effect of turbulence on mixing in the limit. The situation with slow reaction and slow diffusion is also studied in the present work. Here the strong stirring by turbulence before significant reaction occurs necessarily leads to a homogenized limit with the strong mixing effects of turbulence expressed by a rigorous turbulent diffusivity modifying the reaction‐diffusion equations. Physical examples from non‐premixed turbulent combustion and cloud microphysics modeling are utilized throughout the paper to motivate and interpret the mathematical results. © 2000 John Wiley & Sons, Inc.     

Scaling law of diffusivity generated by a noisy telegraph signal with fractal intermittency

In many complex systems the non-linear cooperative dynamics determine the emergence of self-organized, metastable, structures that are associated with a birth–death process of cooperation. This is found to be described by a renewal point process, i.e., a sequence of crucial birth–death events corresponding to transitions among states that are faster than the typical long-life time of the metastable states. Metastable states are highly correlated, but the occurrence of crucial events is typically associated with a fast memory drop, which is the reason for the renewal condition. Consequently, these complex systems display a power-law decay and, thus, a long-range or scale-free behavior, in both time correlations and distribution of inter-event times, i.e., fractal intermittency.The emergence of fractal intermittency is then a signature of complexity. However, the scaling features of complex systems are, in general, affected by the presence of added white or short-term noise. This has been found also for fractal intermittency.In this work, after a brief review on metastability and noise in complex systems, we discuss the emerging paradigm of Temporal Complexity. Then, we propose a model of noisy fractal intermittency, where noise is interpreted as a renewal Poisson process with event rate r_p. We show that the presence of Poisson noise causes the emergence of a normal diffusion scaling in the long-time range of diffusion generated by a telegraph signal driven by noisy fractal intermittency. We analytically derive the scaling law of the long-time normal diffusivity coefficient. We find the surprising result that this long-time normal diffusivity depends not only on the Poisson event rate, but also on the parameters of the complex component of the signal: the power exponent μ of the inter-event time distribution, denoted as complexity index, and the time scale T needed to reach the asymptotic power-law behavior marking the emergence of complexity. In particular, in the range μ < 3, we find the counter-intuitive result that normal diffusivity increases as the Poisson rate decreases.Starting from the diffusivity scaling law here derived, we propose a novel scaling analysis of complex signals being able to estimate both the complexity index μ and the Poisson noise rate r_p.     

Numerical aspects of TV flow

The singular diffusion equation called total variation (TV) flow plays an important role in image processing and appears to be suitable for reducing oscillations in other types of data. Due to its singularity for zero gradients, numerical discretizations have to be chosen with care. We discuss different ways to implement TV flow numerically, and we show that a number of discrete versions of this equation may introduce oscillations such that the scheme is in general not TV-decreasing. On the other hand, we show that TV flow may act self-stabilising: even if the total variation increases by the filtering process, the resulting oscillations remain bounded by a constant that is proportional to the ratio of mesh widths. For our analysis we restrict ourselves to the one-dimensional setting.     

Asympototic Behavior of Semilinear Periodic-Parbolic Boundary Value Problem

51.IntroductionInthispaper,westudythefollowingperiodicboundaryproblemwithNeumannboundaryconditionwhereO=RNisabounddomainwithC1 '(6>o)boundary.LetSuppose'm(x,t)eFandf.,.,$)eFforthe$beIongtoaboundsubsetofR,anditissat-isfiesForthesystem(1.1)',wesupposeitsati…     

On the global regularity of <Emphasis Type="Italic">N</Emphasis>-dimensional generalized Boussinesq system

We study the N-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.     

Convection–diffusion equation with space–time ergodic random flow

We prove the homogenization of convection-diffusion in a time-dependent, ergodic, incompressible random flow which has a bounded stream matrix and a constant mean drift. We also prove two variational formulas for the effective diffusivity. As a consequence, we obtain both upper and lower bounds on the effective diffusivity. Received: 17 December 1996/Revised revision: 9 February 1998     

A Geometric Heat-Flow Theory of Lagrangian Coherent Structures

We consider Lagrangian coherent structures (LCSs) as the boundaries of material subsets whose advective evolution is metastable under weak diffusion. For their detection, we first transform the Eulerian advection–diffusion equation to Lagrangian coordinates, in which it takes the form of a time-dependent diffusion or heat equation. By this coordinate transformation, the reversible effects of advection are separated from the irreversible joint effects of advection and diffusion. In this framework, LCSs express themselves as (boundaries of) metastable sets under the Lagrangian diffusion process. In the case of spatially homogeneous isotropic diffusion, averaging the time-dependent family of Lagrangian diffusion operators yields Froyland’s dynamic Laplacian. In the associated geometric heat equation, the distribution of heat is governed by the dynamically induced intrinsic geometry on the material manifold, to which we refer as the geometry of mixing. We study and visualize this geometry in detail, and discuss connections between geometric features and LCSs viewed as diffusion barriers in two numerical examples. Our approach facilitates the discovery of connections between some prominent methods for coherent structure detection: the dynamic isoperimetry methodology, the variational geometric approaches to elliptic LCSs, a class of graph Laplacian-based methods and the effective diffusivity framework used in physical oceanography.

     

非牛顿幂律流体球向不定常渗流

本文研究了弱压缩非牛顿幂律流体球向不定常渗流,导出了抛物型偏微分非线性方程.球向扩散方程是其特殊情况.用Laplace变换的方法,找到了线性化后方程的解析解和渐近解.用影响半径的概念和平均值方法求得了近似解.渐近解和近似解的结构是相似的,从而丰富了非牛顿流体一维不定常渗流的理论.     

The Asymptotic Limit for the 3D Boussinesq System

In this paper, we show the asymptotic limit for the 3D Boussinesq system with zero viscosity limit or zero diffusivity limit. By the classical energy method, we prove that as viscosity(or diffusivity) coefficient goes to zero the solutions of the fully viscous equations converges to those of zero viscosity(or zero diffusivity) equations, which extend the previous results on the asymptotic limit under the conditions of the zero parameter(zero viscosity ν = 0 or zero diffusivity η = 0) in 2D case separately.     

Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction

In this paper a reaction–diffusion model describing two interacting pioneer and climax species is considered. The role of diffusivity and forcing (stocking or harvesting of the species) on the nonlinear stability of a coexistence equilibrium is analysed. The study is performed in the context of a new approach to nonlinear L²‐stability based on the analysis of stability of the zero solution of a suitable linear system of ordinary differential equations. Theorems concerning the effect of forcing and diffusivity on the dynamics are established and stability–instability thresholds for the system are obtained. An example to illustrate the practical use of the results is also provided. Copyright © 2008 John Wiley & Sons, Ltd.     

A Functional Non-Central Limit Theorem for Jump-Diffusions with Periodic Coefficients Driven by Stable Lévy-Noise

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Lévy-processes with stability index α>1. The limit process turns out to be an α-stable Lévy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.