Investigating Asymptotic Suction Boundary Layers using a One-Dimensional Stochastic Turbulence Model

The one-dimensional turbulence model (ODT) is applied to study turbulent asymptotic suction boundary layers for a Reynolds number of Re = u_∞/v₀ = 333, where u_∞ and v₀ are the free stream and suction velocity, respectively. In here we will demonstrate that a large eddy suppression mechanism may reduce the influence of ODT model parameters, such as the viscous cut-off parameter Z. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Quasi-static Non=Newtonian Fluids with Power-law

We discuss certain classes of quasi-static non-Newtonian fluids for which a power-law of the form σ^D=∇ϕ(ℰv) holds. Here σ^D is the stress deviator, v the velocity field, ℰv its symmetric derivative and ϕ is the function \[ \phi ({\cal E}v)=\frac 12\mu _\infty \left| {\cal E}v\right| ⁁2+\frac 1p\mu _0\left\{ \begin{array}{c} \left( 1+\left| {\cal E}v\right| ⁁2\right) ⁁{p/2} \\ \text{or} \\ \left| {\cal E}v\right| ⁁p \end{array} \right\}, \] ϕ(ℰv)=1 2 μ_∞∣ℰv∣²+1 p μ₀ (1+∣ℰv∣²)^p/2 or ∣ℰv∣^p, μ_∞⩾0, μ₀⩾0, μ_∞+μ₀>0, 1<p<∞. We then prove various regularity results for the velocity field v, for example differentiability almost everywhere and local boundedness of the tensor ℰv.     

The Incompressible Limit and the Initial Layer of the Compressible Euler Equation in ℝn+

We consider the incompressible limit of the compressible Euler equation in the half-space ℝⁿ₊. It is proved that the solutions of the non-dimensionalized compressible Euler equation converge to the solution of the incompressible Euler equation when the Mach number tends to zero. If the initial data v^∞₀ do not satisfy the condition ‘∇⋅v^∞₀=0’, then the initial layer will appear. © 1997 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Holomorphic Dependence of the Heins Map

Let D be a bounded convex domain and Hol _c (D,D) the set of holomorphic maps from D to C ⁿ with image relatively compact in D. Consider Hol _c (D,D) as a open set in the complex Banach space H ^∞ _n (D) of bounded holomorphic maps from D to C ⁿ . We show that the map τ: Hol _c (D,D) → D (called the Heins map for D equals to the unit disc of C) which associates to ? ∈ Hol _c (D,D) its unique fixed point τ? ∈ D is holomorphic and its differential is given by dτ_?(v) = (Id-df_τ(?))^?1 v(τ(?)) for v ∈ H ^∞ _n (D).     

Bifurcation and asymptotic bifurcation for equations involving A-proper mappings with applications to differential equations

Let X, Y be real Banach spaces, T: X → YA-proper, and C: X → Y compact. Section 1 of this paper is devoted to the study of bifurcation and asymptotic bifurcation problems for Eq. $\text{(1): Tx ? λCx = 0}$. In Theorem 1 it is shown that if T(0) = C(0) = 0 and T and C have F-derivatives T₀ and C₀ at 0 with T₀A-proper and injective, then each eigenvalue of T₀x ? λC₀x = 0 of odd multiplicity is a bifurcation point for Eq. (1). Theorem 2 shows that if T and C have asymptotic derivatives T_∞ and C_∞, then each eigenvalue of T_∞x ? λC_∞x = 0 of odd multiplicity is an asymptotic bifurcation point for Eq. (1). Special cases are treated when Y = X and T = I ? F with Fk-ball-contractive or when Y ≠ X and T is either of type (S) or of strongly accretive type. Section 2 is devoted to applications of Theorems 1 and 2 to bifurcation problems involving elliptic operators. The usefulness of Theorems 1 and 2 stems from the fact that they are directly applicable to differential eigenvalue problems without the preliminary reduction of Eq. (1) to equivalent problems involving compact operators. Moreover, in some cases they are applicable in situations to which the known bifurcation results are not applicable.     

Large time behavior of the Lp norm of Schrödinger semigroups

LetV ? 0, V?C₀^∞(R^v) with v ? 3 be such that $\text{H = ?}\text{1}\text{2}\text{Δ + V ? 0}$ but for any $\text{ε > 0, ?}\text{1}\text{2}\text{Δ + (1 + ε)V}$ is not positive. We determine the exact rate of divergence of the norm of e^?tH as a map from L^∞ to L^∞. A number of related problems are discussed.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].     

Topological entropy zero and asymptotic pairs

Let T be a continuous map on a compact metric space (X, d). A pair of distinct points x, y ∈ X is asymptotic if lim _n→∞ d(T ⁿ x, T ⁿ y) = 0. We prove the following four conditions to be equivalent: 1. h _top(T) = 0; 2. (X, T) has a (topological) extension (Y,S) which has no asymptotic pairs; 3. (X, T) has a topological extension (Y ′, S′) via a factor map that collapses all asymptotic pairs; 4. (X, T) has a symbolic extension (i.e., with (Y ′, S′) being a subshift) via a map that collapses asymptotic pairs. The maximal factors (of a given system (X, T)) corresponding to the above properties do not need to coincide.     

Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

A Non-Concentration Estimate for Partially Rectangular Billiards

We consider quasimodes on planar domains with a partially rectangular boundary. We prove that for any ε₀ > 0, an 𝒪(λ^?ε₀ ) quasimode must have L ² mass in the “wings” (in phase space) bounded below by λ^?2?δ for any δ > 0. The proof uses the author's recent work on 0-Gevrey smooth domains to approximate quasimodes on C ^{1, 1} domains. There is an improvement for C ^{k, α} and C ^∞ domains.     

Lower Bounds on the Blow-Up Rate of the Axisymmetric Navier–Stokes Equations II

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in ?³ with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C _? r ^?1+ε |t|^?ε/2 for ? T ₀ ≤ t < 0 and 0 < C _? < ∞ allowed to be large. We prove that v is regular at time zero.     

Stable Blowup Dynamics for the 1‐Corotational Energy Critical Harmonic Heat Flow

We exhibit a stable finite time blowup regime for the 1‐corotational energy critical harmonic heat flow from ?² into a smooth compact revolution surface of ?³ that reduces to the semilinear parabolic problem for a suitable class of functions f. The corresponding initial data can be chosen smooth, well localized, and arbitrarily close to the ground state harmonic map in the energy‐critical topology. We give sharp asymptotics on the corresponding singularity formation that occurs through the concentration of a universal bubble of energy at the speed predicted by van den Berg, Hulshof, and King. Our approach lies in the continuation of the study of the 1‐equivariant energy critical wave map and Schrödinger map with ??² target by Merle, Raphaël, and Rodnianski. © 2012 Wiley Periodicals, Inc.     

On the asymptotic behavior of solutions of nonlinear differential systems

We obtain constructive conditions for the unique solvability of the singular problem dx/dt = f(t, x), x _∞ = 0, where f ∈ C ^(0,1)([0, ∞) × ? ⁿ , ? ⁿ ).     

Laplace-type exact asymptotic formulas for the Bogoliubov Gaussian measure

We obtain new asymptotic formulas for two classes of Laplace-type functional integrals with the Bogoliubov measure. The principal functionals are the L^p functionals with 0 < p < ∞ and two functionals of the exact-upper-bound type. In particular, we prove theorems on the Laplace-type asymptotic behavior for the moments of the L^p norm of the Bogoliubov Gaussian process when the moment order becomes infinitely large. We establish the existence of the threshold value p ₀ = 2+4π ² /β ² ω ₂ , where β > 0 is the inverse temperature and ω > 0 is the harmonic oscillator eigenfrequency. We prove that the asymptotic behavior under investigation differs for 0 < p < p ₀ and p > p ₀ . We obtain similar asymptotic results for large deviations for the Bogoliubov measure. We establish the scaling property of the Bogoliubov process, which allows reducing the number of independent parameters.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Existence and asymptotic behavior to the incompressible nematic liquid crystal flow in the whole space

In this article, first of all, the global existence and asymptotic stability of solutions to the incompressible nematic liquid crystal flow is investigated when initial data are a small perturbation near the constant steady state (0,δ₀); here, δ₀ is a constant vector with |δ₀|=1. Precisely, we show the existence and asymptotic stability with small initial data for . The initial data class of us is not entirely included in the space BMO^?1×BMO and contains strongly singular functions and measures. As an application, we obtain a class of asymptotic existence of a basin of attraction for each self‐similar solution with homogeneous initial data. We also study global existence of a large class of decaying solutions and construct an explicit asymptotic formula for ∣x∣→∞, relating the self‐similar profile (U(x),D(x)) to its corresponding initial data (u₀,d₀). In two dimensions, we obtain higher‐order asymptotics of (u(x),d(x)). Copyright © 2016 John Wiley & Sons, Ltd.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Tame comodule type,roiter bocses,and a geometry context for coalgebras

We study the class of coalgebras C of fc-tame comodule type introduced by the author. With any basic computable K-coalgebra C and a bipartite vector v = (v′|v″) ∈ K ₀(C) × K ₀(C), we associate a bimodule matrix problem Mat ^v _C (ℍ), an additive Roiter bocs B ^C _v , an affine algebraic K-variety Comod ^C _v , and an algebraic group action G ^C _v × Comod ^C _v → Comod ^C _v . We study the fc-tame comodule type and the fc-wild comodule type of C by means of Mat ^v _C (ℍ), the category rep _K (B ^C _v ) of K-linear representations of B ^C _v , and geometry of G ^C _v -orbits of Comod ^C _v . For computable coalgebras C over an algebraically closed field K, we give an alternative proof of the fc-tame-wild dichotomy theorem. A characterization of fc-tameness of C is given in terms of geometry of G ^C _v -orbits of Comod _v . In particular, we show that C is fc-tame of discrete comodule type if and only if the number of G ^C _v -orbits in Comod ^C _v is finite for every v = (v′|v″) ∈ K ₀(C) × K ₀(C).     

On the asymptotic tensor C*-norm

We introduce a new asymptotic one-sided and symmetric tensor norm, the latter of which can be considered as the minimal tensor norm on the category of separable C^*-algebras with homotopy classes of asymptotic homomorphisms as morphisms. We show that the one-sided asymptotic tensor norm differs in general from both the minimal and the maximal tensor norms and discuss its relation to semi-invertibility of C^*-extensions. Received: 23 September 2004; revised: 30 May 2005     

Correlation between pole location and asymptotic behavior for Painlevé I solutions

We extend the technique of asymptotic series matching to exponential asymptotics expansions (transseries) and show by using asymptotic information that the extension provides a method of finding singularities of solutions of nonlinear differential equations. This transasymptotic matching method is applied to Painlevé's first equation, P1. The solutions of P1 that are bounded in some direction towards infinity can be expressed as series of functions obtained by generalized Borel summation of formal transseries solutions; the series converge in a neighborhood of infinity. We prove (under certain restrictions) that the boundary of the region of convergence contains actual poles of the associated solution. As a consequence, the position of these exterior poles is derived from asymptotic data. In particular, we prove that the location of the outermost pole x_p(C) on ℝ⁺ of a solution is monotonic in a parameter C describing its asymptotics on anti‐Stokes lines and obtain rigorous bounds for x_p(C). We also derive the behavior of x_p(C) for large C ∈ ℂ. The appendix gives a detailed classical proof that the only singularities of solutions of P1 are poles. © 1999 John Wiley & Sons, Inc.