On generalized energy equality of the Navier-Stokes equations

We show that for every initial dataa εL ²(Ω) there exists a weak solutionu of the Navier-Stokes equations satisfying the generalized energy inequality introduced by Caffarelli-Kohn-Nirenberg forn=3. We also show that if a weak solutionu εL ^s (0,T;L ^q (Ω)) with 2/q + 2/s ≤ 1 and 3/q + 1/s ≤ 1 forn=3, or 2/q + 2/s ≤ 1 andq ≥ 4 forn ≥ 4, thenu satisfies both the generalized and the usual energy equalities. Moreover we show that the generalized energy equality holds only under the local hypothesis thatu εL ^s (ε, T; L ^q (K)) for all compact setsK ⊂ ⊂ Ω and all 0 <ε <T with the same (q, s) as above when 3 ≤n ≤ 10.     

Existence of maximal solutions

Let Ω be a plane bounded region. Let U = {U_μ(P):μ(P)εL∞(Ω), u_μ ε H²_{2, 0}(Ω) and a(P, μ(P))u_μ,xx + 2b(P, μ(P))u_μ,xy + c(P, μ(P))u_μ,vv = ƒ(P) for P ε Ω; here we are given a(P, X), b(P, X), c(P, X) ε L_∞(Ω × E¹), ƒ(P) ε L_p(Ω) with p > 2, and our partial differential equation is uniformly elliptic. The functions μ(P) are called profiles. We establish sufficient conditions—which when they apply are constructive—that there exist a μ₀ ε L_∞(Ω) such that u_μ0 (P) u_μ(P) for all P ε Ω and for each μ ε L_∞(Ω). Similar results are obtained for a difference equation and convergence is proved.     

A generalized mean value inequality for subharmonic functions

If u ≥ 0 is subharmonic on a domain Ω in ⁿ and 0 < p < 1, then it is well-known that there is a constant C(n,p) ≥ 1 such u(x)^p ≤ C)n,p) MV )u^p,B(x,r)) for each ball B(x,r)) Ω. We show more generally that a similar result holds for functions ψ : ₊ → ₊ may be any surjective, concave function whose inverse ψ⁻¹ satisfies the Δ₂-condition.     

Global existence and blow-up of solutionsfor a higher-order kirchhoff-type equation with nonlinear dissipation

This paper deals with the higher-order Kirchhoff-type equation with nonlinear dissipationu_tt+(∫_Ω׀D^mu׀²dx)^q(−Δ)^mu+u_t׀u_t׀^r=׀u׀^pu,xΩ,t>0,in a bounded domain, where m < 1 is a positive integer, q, p, r < 0 arepositive constants. We obtain that the solution exists globally if p ≤ r, while ifp > max r, 2q , then for any initial data with negative initial energy, the solution blowsup at finite time in L^p+2 norm.     

Asymptotiques d’un systeme dynamique conservatif associe a une equation elliptique non lineaire

LetM be a compact riemannian manifold,h an odd function such thath(r)/r is non-decreasing with limit 0 at 0. Letf(r)=h(r)-γr and assume there exist non-negative constantsA andB and a realp>1 such thatf(r)>Ar ^P-B. We prove that any non-negative solutionu ofu _tt+Δ_gu=f(u) onM x ℝ⁺ satisfying Dirichlet or Neumann boundary conditions on ϖM converges to a (stationary) solution of Δ _g Φ=f(Φ) onM with exponential decay of ‖u-Φ‖C ²(M). For solutions with non-constant sign, we prove an homogenisation result for sufficiently small λ; further, we show that for every λ the map (u(0,·),u _t(0,·))→(u(t,·), u _t(t,·)) defines a dynamical system onW ^1/2(M)⊂C(M)×L ²(M) which possesses a compact maximal attractor.      

Norms Concerning Subdivision Sequences and Their Applications in Wavelets

Let a:=(a(α))_{α ^s} be a finitely supported sequence of r×r matrices and M be a dilation matrix. The subdivision sequence {(a_n(α))_{α ^s}:n } is defined by a₁=a and

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Let 1≤p≤∞ and f=(f¹,…,f^r)^T be a vector of compactly supported functions in L_p( ^s). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L_p-norms of the combinations of the shifts of f with the subdivision sequence coefficients: Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method.     

On the structure of biharmonic functions satisfying the clamped plate conditions on a right angle

Let u(r,θ) be biharmonic and bounded in the circular sector ¦θ¦ < π/4, 0 < r < ρ (ρ > 1) and vanish together with δu/δθ when ¦θ¦ = π/4. We consider the transform û(p,θ) = ∝₀¹r^{p − 1}u(r,θ)dr. We show that for any fixed θ₀ u(p,θ₀) is meromorphic with no real poles and cannot be entire unless u(r, θ₀) ≡ 0. It follows then from a theorem of Doetsch that u(r, θ₀) either vanishes identically or oscillates as r → 0.     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Existence of three solutions for a class of elliptic eigenvalue problems

In this paper, we consider a problem of the type −Δu = λ(f(u) + μg(u)) in Ω, u¦∂Ω = 0, where Ω Rⁿ is an open-bounded set, f, g are continuous real functions on R, and λ, μ ε R. As an application of a new approach to nonlinear eigenvalues problems, we prove that, under suitable hypotheses, if ¦μ¦ is small enough, then there is some λ > 0 such that the above problem has at least three distinct weak solutions in W₀^1,2(Ω).     

The Pointwise Densities of the Cantor Measure

Let be the classical middle-third Cantor set and let μ be the Cantor measure. Set s = log 2/log 3. We will determine by an explicit formula for every point x the upper and lower s-densities Θ*^s(μ , x), Θ*^s(μ , x) of the Cantor measure at the point x, in terms of the 3-adic expansion of x. We show that there exists a countable set F such that 9(Θ*^s(μ , x))^{− 1/s} + (Θ*^s(μ , x))^{− 1/s} = 16 holds for x \F. Furthermore, for μ_C almost all x, Θ*^s(μ , X) − 2 · 4^{− s} and Θ*^s(μ , x) = 4^{− s}. As an application, we will show that the s-dimensional packing measure of the middle-third Cantor set is 4^s.     

On piecewise-polynomial approximation of functions with a bounded fractional derivative in an Lp-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε [0, 1), and ƒ^{(r + α)} is the classical fractional derivative, i.e., ƒ^{(r + α)}(y) = ∝₀¹, ^α d(ƒ^(r)(t)). We prove that, for any such function ƒ, there exists a piecewise-polynomial of degree s that interpolates ƒ at n equally spaced points and that approximates ƒ with an error (in sup-norm) ƒ^{(r + α)}_p O(n^{−(r+α−1/p}). We also prove that no algorithm based on n function and/or derivative values of ƒ has the error equal ƒ^{(r + α)}_p O(n^{−(r+α−1/p}) for any ƒ. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s r. Hence, even without knowing the actual regularity (r, α, and p) of ƒ, we can approximate the function ƒ with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.     

Theorems of large deviations in the multivariate invariance principle

Let B be a real separable Banach space with norm |ß|_B, X, X₁, X₂, … be a sequence of centered independent identically distributed random variables taking values in B. Let s_n = s_n(t), 0 ≤ t ≤ 1 be the random broken line such that s_n(0) = 0, s_n(k/n) = n^−1/2 Σ_i=1^k X_i for n = 1, 2, … and k = 1, …, n. Denote |s_n|_B = sup_{0 ≤ t ≤ 1} |s_n(t)|_B and assume that w(t), 0 ≤ t ≤ 1 is the Wiener process such that covariances of w(1) and X are equal. We show that under appropriate conditions P(|s_n|_B > r) = P(|w|_B > r)(1 + o(1)) and give estimates of the remainder term. The results are new already in the case of B having finite dimension.     

On the asymptotic behaviour of a semigroup of linear operators

Let T = {T(t)}_{t ≥ 0} be a C₀-semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω_Lp(T) of weak p-integrability of T (1 ≤ p < ∞), the abscissa ω^p_R(A) of p-boundedness of the resolvent of the generator A of T (1 ≤ p ≤ ∞), and the growth bounds ω_β(T), β ≥ 0, of T. Our main results are as follows.

1. (i) Let T be a C₀-semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω_{1 − ε}(T) < 0 for some ε > 0.

2. (ii) Let T be a C₀-semigroup on L^p such that the resolvent of the generator is uniformly bounded in the right half plane. Then ω_β(T) < 0 for all β>¦1/p − 1/p′¦, 1/p + 1/p′ = 1.

3. (iii) Let 1 ≤ p ≤ 2 and let T be a weakly L^p-stable C₀-semigroup on a Banach space X. Then for all β>1/p we have ω_β(T) ≤ 0.

Further, we give sufficient conditions in terms of ω^q_R(A) for the existence of L^p-solutions and W^1,p-solutions (1 ≤ p ≤ ∞) of the abstract Cauchy problem for a general class of operators A on X.     

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Algebraic Approaches to Periodic Arithmetical Maps

A residue class a + n with weight λ is denoted by λ, a, n. For a finite system = {λ_s, a_s, n_s}^k_{s = 1} of such triples, the periodic map w (x) = ∑_{ns|x − as} λ_s is called the covering map of . Some interesting identities for those with a fixed covering map have been known; in this paper we mainly determine all those functions f : Ω → such that ∑^k_{s = 1} λ_sf(a_s + n_s ) depends only on w where Ω denotes the family of all residue classes. We also study algebraic structures related to such maps f, and periods of arithmetical functions ψ(x) = ∑^k_{s = 1} λ_se^2πia_sx/n_s and ω(x) = |{1 ≤ s ≤ k : (x + a_s, n_s) = 1}|.     

The Complexity of Indefinite Elliptic Problems with Noisy Data

We study the complexity of second-order indefinite elliptic problems −div(au) +bu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, the error being measured in theH¹(Ω)-norm. The problem elementsfbelong to the unit ball ofW^{r, p}, (Ω), wherep [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations off,a, orb(or their derivatives). The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton^−r/d+ δ and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the -complexity (minimal cost of calculating an -approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ^−s(fors> 0), then the complexity is proportional to (1/)^{d/r + s}.     

Periodic and Homoclinic Solutions of Extended Fisher–Kolmogorov Equations

In this paper we study the existence of periodic solutions of the fourth-order equations u^iv − pu″ − a(x)u + b(x)u³ = 0 and u^iv − pu″ + a(x)u − b(x)u³ = 0, where p is a positive constant, and a(x) and b(x) are continuous positive 2L-periodic functions. The boundary value problems (P₁) and (P₂) for these equations are considered respectively with the boundary conditions u(0) = u(L) = u″(0) = u″(L) = 0. Existence of nontrivial solutions for (P₁) is proved using a minimization theorem and a multiplicity result using Clark's theorem. Existence of nontrivial solutions for (P₂) is proved using the symmetric mountain-pass theorem. We study also the homoclinic solutions for the fourth-order equation u^iv + pu″ + a(x)u − b(x)u² − c(x)u³ = 0, where p is a constant, and a(x), b(x), and c(x) are periodic functions. The mountain-pass theorem of Brezis and Nirenberg and concentration-compactness arguments are used.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

Asymptotic behaviour of nonlinear parabolic equations with critical exponents. A dynamical systems approach

We investigate the large-time behaviour of solutions to the nonlinear heat-conduction equation with absorption u_t = Δ(u^{σ + 1}) − u^β in Q = R^N × (0, ∞) (E) with N 1, σ > 0 and critical absorption exponent β = σ + 1 + 2/N; the initial function u(x, 0) = 0 is assumed to be integrable, nonnegative and compactly supported. We prove that u converges as t → ∞ to a unique self-similar function which is a contracted version of one of the asymptotic profiles of the nonabsorptive problem u_t = Δ(u^{σ + 1}), the same for any initial data. The cornerstone of the proof is a result about ω-limits of (infinite-dimensional) asymptotical dynamical systems. Combining this result with an asymptotic evaluation of the mass function as well as typical PDE estimates gives the behaviour of (E) for large times.Similar unusual asymptotic behaviour is obtained for the equation u_t = div(¦Du¦^σ Du) − u^β with same conditions on σ and u(x, 0) and critical value for β = σ + 1 + (σ + 2)/N.     

Uniform Estimates for a Class of Evolution Equations

L^∞ estimates are derived for the oscillatory integral ∫⁺₀^∞e^{−i(xλ + (1/m) tλm)}a(λ) dλ, where 2 ≤ m and (x, t) × ₊. The amplitude a(λ) can be oscillatory, e.g., a(λ) = e^{it (λ)} with (λ) a polynomial of degree ≤ m − 1, or it can be of polynomial type, e.g., a(λ) = (1 + λ)^k with 0 ≤ k ≤ (m − 2). The estimates are applied to the study of solutions of certain linear pseudodifferential equations, of the generalized Schrödinger or Airy type, and of associated semilinear equations.