Non-autonomous 2D Navier–Stokes System with Singularly Oscillating External Force and its Global Attractor

We study the global attractor of the non-autonomous 2D Navier–Stokes (N.–S.) system with singularly oscillating external force of the form . If the functions g ₀(x, t) and g ₁ (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor is bounded in the space H, however, its norm may be unbounded as since the magnitude of the external force is growing. Assuming that the function g ₁ (z, t) has a divergence representation of the form where the functions (see Section 3), we prove that the global attractors of the N.–S. equations are uniformly bounded with respect to for all . We also consider the “limiting” 2D N.–S. system with external force g ₀(x, t). We have found an estimate for the deviation of a solution of the original N.–S. system from a solution u ⁰(x, t) of the “limiting” N.–S. system with the same initial data. If the function g ₁ (z, t) admits the divergence representation, the functions g ₀(x, t) and g ₁ (z, t) are translation compact in the corresponding spaces, and , then we prove that the global attractors converges to the global attractor of the “limiting” system as in the norm of H. In the last section, we present an estimate for the Hausdorff deviation of from of the form: in the case, when the global attractor is exponential (the Grashof number of the “limiting” 2D N.–S. system is small).      

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equation

Fine Properties of Self-Similar Solutions of the Navier–Stokes Equations

We study the solutions of the nonstationary incompressible Navier–Stokes equations in , of self-similar form , obtained from small and homogeneous initial data a(x). We construct an explicit asymptotic formula relating the self-similar profile U(x) of the velocity field to its corresponding initial datum a(x).     

The Asymptotic Finite-dimensional Character of a Spectrally-hyperviscous Model of 3D Turbulent Flow

We obtain attractor and inertial-manifold results for a class of 3D turbulent flow models on a periodic spatial domain in which hyperviscous terms are added spectrally to the standard incompressible Navier–Stokes equations (NSE). Let P _m be the projection onto the first m eigenspaces of A =−Δ, let μ and α be positive constants with α ≥3/2, and let Q _m =I − P _m , then we add to the NSE operators μ A _φ in a general family such that A _φ≥Q _m A ^α in the sense of quadratic forms. The models are motivated by characteristics of spectral eddy-viscosity (SEV) and spectral vanishing viscosity (SVV) models. A distinguished class of our models adds extra hyperviscosity terms only to high wavenumbers past a cutoff λ _m0 where m ₀ ≤ m, so that for large enough m ₀ the inertial-range wavenumbers see only standard NSE viscosity. We first obtain estimates on the Hausdorff and fractal dimensions of the attractor (respectively and ). For a constant K _α on the order of unity we show if μ ≥ ν that and if μ ≤ ν that where ν is the standard viscosity coefficient, l ₀ = λ₁^−1/2 represents characteristic macroscopic length, and is the Kolmogorov length scale, i.e. where is Kolmogorov’s mean rate of dissipation of energy in turbulent flow. All bracketed constants and K _α are dimensionless and scale-invariant. The estimate grows in m due to the term λ _m /λ₁ but at a rate lower than m ^3/5, and the estimate grows in μ as the relative size of ν to μ. The exponent on is significantly less than the Landau–Lifschitz predicted value of 3. If we impose the condition , the estimates become for μ ≥ ν and for μ ≤ ν. This result holds independently of α, with K _α and c _α independent of m. In an SVV example μ ≥ ν, and for μ ≤ ν aspects of SEV theory and observation suggest setting for 1/c within α orders of magnitude of unity, giving the estimate where c _α is within an order of magnitude of unity. These choices give straight-up or nearly straight-up agreement with the Landau–Lifschitz predictions for the number of degrees of freedom in 3D turbulent flow with m so large that (e.g. in the distinguished-class case for m ₀ large enough) we would expect our solutions to be very good if not virtually indistinguishable approximants to standard NSE solutions. We would expect lower choices of λ _m (e.g. with a > 1) to still give good NSE approximation with lower powers on l ₀/l _ε, showing the potential of the model to reduce the number of degrees of freedom needed in practical simulations. For the choice , motivated by the Chapman–Enskog expansion in the case m = 0, the condition becomes , giving agreement with Landau–Lifschitz for smaller values of λ _m then as above but still large enough to suggest good NSE approximation. Our final results establish the existence of a inertial manifold for reasonably wide classes of the above models using the Foias/Sell/Temam theory. The first of these results obtains such an of dimension N > m for the general class of operators A _φ if α > 5/2. The special class of A _φ such that P _m A _φ = 0 and Q _m A _φ ≥ Q _m A ^α has a unique spectral-gap property which we can use whenever α ≥ 3/2 to show that we have an inertial manifold of dimension m if m is large enough. As a corollary, for most of the cases of the operators A _φ in the distinguished-class case that we expect will be typically used in practice we also obtain an , now of dimension m ₀ for m ₀ large enough, though under conditions requiring generally larger m ₀ than the m in the special class. In both cases, for large enough m (respectively m ₀), we have an inertial manifold for a system in which the inertial range essentially behaves according to standard NSE physics, and in particular trajectories on are controlled by essentially NSE dynamics.      

Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ₁ is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.      

Uniqueness and Nonuniqueness of Viscosity Solutions to Aronsson’s Equation

For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:

A New Approach to the Fundamental Theorem of Surface Theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a _αβ ) of order two and a field of symmetric matrices (b _αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of , then there exists an immersion such that these fields are the first and second fundamental forms of the surface , and this surface is unique up to proper isometries in . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a _αβ and b _αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation

Resolvent Estimates and Maximal Regularity in Weighted L q -spaces of the Stokes Operator in an Infinite Cylinder

Let be an infinite cylinder of , n ≥ 3, with a bounded cross-section of C ^1,1-class. We study resolvent estimates and maximal regularity of the Stokes operator in for 1 < q, r < ∞ and for arbitrary Muckenhoupt weights ω ∈ A _r with respect to x′ ∈ Σ. The proofs use an operator-valued Fourier multiplier theorem and techniques of unconditional Schauder decompositions based on the -boundedness of the family of solution operators for a system in Σ parametrized by the phase variable of the one-dimensional partial Fourier transform. Supported by the Gottlieb Daimler- und Karl Benz-Stiftung, grant no. S025/02-10/03.     

Uniform Convergence for Approximate Traveling Waves in Linear Reaction–Diffusion–Hyperbolic Systems

In this paper we study linear reaction–hyperbolic systems of the form , (i = 1, 2, ..., n) for x > 0, t > 0 coupled to a diffusion equation for p ₀ = p ₀(x, y, θ, t) with “near-equilibrium” initial and boundary data. This problem arises in a model of transport of neurofilaments in axons. The matrix (k _ij ) is assumed to have a unique null vector with positive components summed to 1 and the v _j are arbitrary velocities such that . We prove that as the solution converges to a traveling wave with velocity v and a spreading front, and that the convergence rate in the uniform norm is , for any small positive α.     

Spatial and Dynamical Chaos Generated by Reaction–Diffusion Systems in Unbounded Domains

We consider in this article a nonlinear reaction–diffusion system with a transport term (L,∇ _x )u, where L is a given vector field, in an unbounded domain Ω. We prove that, under natural assumptions, this system possesses a locally compact attractor in the corresponding phase space. Since the dimension of this attractor is usually infinite, we study its Kolmogorov’s ɛ-entropy and obtain upper and lower bounds of this entropy. Moreover, we give a more detailed study of the spatio-temporal chaos generated by the spatially homogeneous RDS in . In order to describe this chaos, we introduce an extended (n + 1)-parametrical semigroup, generated on the attractor by 1-parametrical temporal dynamics and by n-parametrical group of spatial shifts ( = spatial dynamics). We prove that this extended semigroup has finite topological entropy, in contrast to the case of purely temporal or purely spatial dynamics, where the topological entropy is infinite. We also modify the concept of topological entropy in such a way that the modified one is finite and strictly positive, in particular for purely temporal and for purely spatial dynamics on the attractor. In order to clarify the nature of the spatial and temporal chaos on the attractor, we use (following Zelik, 2003, Comm. Pure. Appl. Math. 56(5), 584–637) another model dynamical system, which is an adaptation of Bernoulli shifts to the case of infinite entropy and construct homeomorphic embeddings of it into the spatial and temporal dynamics on . As a corollary of the obtained embeddings, we finally prove that every finite dimensional dynamics can be realized (up to a homeomorphism) by restricting the temporal dynamics to the appropriate invariant subset of .     

The Semigeostrophic Equations Discretized in Reference and Dual Variables

We study the evolution of a system of n particles in . That system is a conservative system with a Hamiltonian of the form , where W ₂ is the Wasserstein distance and μ is a discrete measure concentrated on the set . Typically, μ(0) is a discrete measure approximating an initial L ^∞ density and can be chosen randomly. When d  =  1, our results prove convergence of the discrete system to a variant of the semigeostrophic equations. We obtain that the limiting densities are absolutely continuous with respect to the Lebesgue measure. When converges to a measure concentrated on a special d–dimensional set, we obtain the Vlasov–Monge–Ampère (VMA) system. When, d = 1 the VMA system coincides with the standard Vlasov–Poisson system.     

2D Stochastic Navier–Stokes Equations with a Time-Periodic Forcing Term

We study the long time behavior of the solution X(t, s, x) of a 2D-Navier–Stokes equation subjected to a periodic time dependent forcing term. We prove in particular that as , approaches a periodic orbit independently of s and x for any continuous and bounded real function .      

Uniform Exponential Dichotomy and Continuity of Attractors for Singularly Perturbed Damped Wave Equations

For , we consider a family of damped wave equations , where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L ²(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A _η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0⁺.     

Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Heat Flow of Harmonic Maps Whose Gradients Belong to L^{n}_{x}L^{\infty}_{t}

For any compact n-dimensional Riemannian manifold (M, g) without boundary, a compact Riemannian manifold without boundary, and 0 < T ≦ +∞, we prove that for n ≧ 4, if u : M × (0, T] → N is a weak solution to the heat flow of harmonic maps such that , then u ∈C ^∞(M × (0, T], N). As a consequence, we show that for n ≧3, if 0 < T < +∞ is the maximal time interval for the unique smooth solution u ∈C ^∞(M × [0, T), N) of (1.1), then blows up as t ↑ T.     

Higher Order Entropies

Higher order entropies are kinetic entropy estimators for fluid models. These quantities are quadratic in the velocity and temperature derivatives and have temperature dependent coefficients. We investigate governing equations for higher order entropies and related a priori estimates in the natural situation where viscosity and thermal conductivity depend on temperature. We establish positivity of higher order derivative source terms in these governing equations provided that is small enough. The temperature factors renormalizing temperature and velocity derivatives then yield majorization of lower order convective terms only when the temperature dependence of transport coefficients is taken into account according to the kinetic theory. In this situation, we obtain entropic principles for higher order entropies of arbitrary order. As an application, we investigate a priori estimates and global existence of solutions when the initial values log(T ₀/T _∞) and are small enough in appropriate spaces.     

Persistence of Semi-Trajectories

We consider diffeomorphisms f of a smooth compact riemannian mainfold M and its suspension flow . Assuming some regularity of the stable (unstable) sets at the points we prove the persistence in the future of {f ⁿ (x), n ≥ 0} or , i.e., that C ⁰ small perturbations g of f have a semi-trajectory that closely shadows {f ⁿ (x), n ≥ 0} and that the suspension of g has also a semi-trajectory that closely shadows . In case x belongs to a minimal set of f we show that the assumptions concerning the regularity of stable and unstable sets could be reduced to a neighbourhood of x.     

Control for going from hovering to small speed flight of a model insect

The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle （δФ） results in a horizontal forward or backward flight; a change in the stroke amplitude （δФ） or a equal change in the down- and upstroke angles of attack （δα1） results in a vertical climb or decent; a proper combination of δФ and δФ controls （or δФ and δα1 controls） can give a flight of any （small） speed in any desired direction.