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).      

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:

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.     

The Boundary Riemann Solver Coming from the Real Vanishing Viscosity Approximation

We study the limit of the hyperbolic–parabolic approximation

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

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.     

Crack Initiation in Brittle Materials

In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → t g(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time t _i > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional where , k > 0 and f is a suitable Carathéodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with small enough are such that .     

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.      

Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .     

Existence Theory by Front Tracking for General Nonlinear Hyperbolic Systems

We consider the Cauchy problem for a strictly hyperbolic, N × N quasilinear system in one-space dimension

Existence of a Solution “in the Large” for Ocean Dynamics Equations

For the system of equations describing the large-scale ocean dynamics, an existence and uniqueness theorem is proved “in the large”. This system is obtained from the 3D Navier–Stokes equations by changing the equation for the vertical velocity component u ₃ under the assumption of smallness of a domain in z-direction, and a nonlinear equation for the density function ρ is added. More precisely, it is proved that for an arbitrary time interval [0, T], any viscosity coefficients and any initial conditions

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.     

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     

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⁺.     

Least Supersolution Approach to Regularizing Free Boundary Problems

In this paper, we study a free boundary problem obtained as a limit as ε → 0 to the following regularizing family of semilinear equations , where β _ε approximates the Dirac delta in the origin and F is a Lipschitz function bounded away from 0 and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces that are uniform in ε. This allows to prove that the free boundary of a limit has the “right” weak geometry, in the measure theoretical sense. By the construction of some barriers with curvature, the classification of global profiles of the blow-up analysis is carried out and the limit functions are proven to be viscosity and pointwise solution ( almost everywhere) to a free boundary problem. Finally, the free boundary is proven to be a C ^1,α surface around almost everywhere point. An erratum to this article can be found at     

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.     

On Dual Configurational Forces

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,