Stability of Cahn‐Hilliard fronts

We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. We prove stability of the kink solution of the Cahn‐Hilliard equation ∂_tu = ∂( ∂u − u/2 + u³/2), x ∈ ℝ. The proof is based on an inductive renormalization group method, and we obtain detailed asymptotics of the solution as t → ∞. © 1999 John Wiley & Sons, Inc.     

A quantitative compactness estimate for scalar conservation laws

In the case of a scalar conservation law with convex flux in space dimension one, P. D. Lax proved [Comm. Pure and Appl. Math. 7 (1954)] that the semigroup defining the entropy solution is compact in L for each positive time. The present note gives an estimate of the ?‐entropy in L of the set of entropy solutions at time t > 0 whose initial data run through a bounded set in L¹. © 2005 Wiley Periodicals, Inc.     

The red‐shift effect and radiation decay on black hole spacetimes

We consider solutions to the linear wave equation □_g? = 0 on a (maximally extended) Schwarzschild spacetime with parameter M > 0, evolving from sufficiently regular initial data prescribed on a complete Cauchy surface Σ, where the data are assumed only to decay suitably at spatial infinity. (In particular, the support of ? may contain the bifurcate event horizon.) It is shown that the energy flux F(??) of the solution (as measured by a strictly timelike T? that asymptotically matches the static Killing field) through arbitrary achronal subsets ?? of the black hole exterior region satisfies the bound F(??) ≤ C E(v + u), where v and u denote the infimum of the Eddington‐Finkelstein advanced and retarded time of ??, v₊ denotes max{1, v}, and u₊ denotes max{1, u}, where C is a constant depending only on the parameter M, and E depends on a suitable norm of the solution on the hypersurface t ? u + v = 1. (The bound applies in particular to subsets ?? of the event horizon or null infinity.) It is also shown that ? satisfies the pointwise decay estimate |?| ≤ C Ev in the entire exterior region, and the estimates |r?| ≤ C_R?E(1 + |u|)^?1/2 and |r^1/2?| ≤ C_R?Eu in the region {r ≥ R?} ∩ J⁺(Σ) for any R? > 2M. The estimates near the event horizon exploit an integral energy identity normalized to local observers. This estimate can be thought to quantify the celebrated red‐shift effect. The results in particular give an independent proof of the classical result |?| ≥ C E of Kay and Wald without recourse to the discrete isometries of spacetime. © 2009 Wiley Periodicals, Inc.     

Highly oscillatory multidimensional shocks

We construct geometric optics expansions of high order for oscillatory multidimensional shocks and then show that the expansions are close to exact shock solutions for small wavelengths. Expansions are constructed both for S_ϵ, the oscillatory function defining the shock surface S_ϵ, and for u, the solutions on each side of S_ϵ. The profile equations yield detailed information on the evolution of (u, ψ_ϵ), showing, for example, how new interior oscillations are produced by a variety of shock—interior and interior—interior interactions. A generic small divisor property, L²‐estimates for linearized shock problems with merely Lipschitz coefficients, and a continuation principle based on an unusual Gagliardo‐Nirenberg inequality all play a role in the proofs. © 1999 John Wiley & Sons, Inc.     

On the density of 2‐colorable 3‐graphs in which any four points span at most two edges

Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)Let ex₂(n, K) be the maximum number of edges in a 2‐colorable K‐free 3‐graph (where K={123, 124, 134} ). The 2‐chromatic Turán density of K is $\pi_{2}({K}_{4}^-) =lim_{{n}\to \infty} {ex}_{2}({n}, {K}_{4}^-)/\left(_{3}^{n}\right)$. We improve the previously best known lower and upper bounds of 0.25682 and 3/10?ε, respectively, by showing that This implies the following new upper bound for the Turán density of K In order to establish these results we use a combination of the properties of computer‐generated extremal 3‐graphs for small n and an argument based on “super‐saturation”. Our computer results determine the exact values of ex(n, K) for n≤19 and ex₂(n, K) for n≤17, as well as the sets of extremal 3‐graphs for those n. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 105–114, 2010     

Global attractors for 2‐D wave equations with displacement‐dependent damping

In this paper the long‐time behaviour of the solutions of 2‐D wave equation with a damping coefficient depending on the displacement is studied. It is shown that the semigroup generated by this equation possesses a global attractor in H(Ω) × L₂(Ω) and H²(Ω)∩H(Ω) × H(Ω). Copyright © 2009 John Wiley & Sons, Ltd.     

Iteratively reweighted least squares minimization for sparse recovery

Under certain conditions (known as the restricted isometry property, or RIP) on the m × N matrix Φ (where m < N), vectors x ∈ ?^N that are sparse (i.e., have most of their entries equal to 0) can be recovered exactly from y := Φx even though Φ^?1(y) is typically an (N ? m)—dimensional hyperplane; in addition, x is then equal to the element in Φ^?1(y) of minimal ??₁‐norm. This minimal element can be identified via linear programming algorithms. We study an alternative method of determining x, as the limit of an iteratively reweighted least squares (IRLS) algorithm. The main step of this IRLS finds, for a given weight vector w, the element in Φ^?1(y) with smallest ??₂(w)‐norm. If x⁽ⁿ⁾ is the solution at iteration step n, then the new weight w⁽ⁿ⁾ is defined by w := [|x|² + ε]^?1/2, i = 1, …, N, for a decreasing sequence of adaptively defined ε_n; this updated weight is then used to obtain x^{(n + 1)} and the process is repeated. We prove that when Φ satisfies the RIP conditions, the sequence x⁽ⁿ⁾ converges for all y, regardless of whether Φ^?1(y) contains a sparse vector. If there is a sparse vector in Φ^?1(y), then the limit is this sparse vector, and when x⁽ⁿ⁾ is sufficiently close to the limit, the remaining steps of the algorithm converge exponentially fast (linear convergence in the terminology of numerical optimization). The same algorithm with the “heavier” weight w = [|x|² + ε]^?1+τ/2, i = 1, …, N, where 0 < τ < 1, can recover sparse solutions as well; more importantly, we show its local convergence is superlinear and approaches a quadratic rate for τ approaching 0. © 2009 Wiley Periodicals, Inc.     

Longtime behavior for a nonlinear wave equation arising in elasto‐plastic flow

The paper studies the longtime behavior of solutions to the initial boundary value problem (IBVP) for a nonlinear wave equation arising in elasto‐plastic flow u_tt?div{|?u|^m?1?u}?λΔu_t+Δ²u+g(u)=f(x). It proves that under rather mild conditions, the dynamical system associated with above‐mentioned IBVP possesses a global attractor, which is connected and has finite Hausdorff and fractal dimension in the phase spaces X₁=H(Ω) × L²(Ω) and X=(H³(Ω)∩H(Ω)) × H(Ω), respectively. Copyright © 2008 John Wiley & Sons, Ltd.     

Graph minors and linkages

Bollobás and Thomason showed that every 22k‐connected graph is k‐linked. Their result used a dense graph minor. In this paper, we investigate the ties between small graph minors and linkages. In particular, we show that a 6‐connected graph with a K minor is 3‐linked. Further, we show that a 7‐connected graph with a K minor is (2,5)‐linked. Finally, we show that a graph of order n and size at least 7n?29 contains a K minor. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 75–91, 2005     

Multi‐to One‐Dimensional Optimal Transport

We consider the Monge‐Kantorovich problem of transporting a probability density on to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level‐set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples. More specifically, fix probability densities f and g on open sets and with . Consider transporting f onto g so as to minimize the cost . We give a nondegeneracy condition on that ensures the set of x paired with [g‐a.e.] y ∈ Y lie in a codimension‐n submanifold of X. Specializing to the case m > n = 1, we discover a nestedness criterion relating s to (f,g) that allows us to construct a unique optimal solution in the form of a map . When and g and f are bounded, the Kantorovich dual potentials (u,υ) satisfy , and the normal velocity V of with respect to changes in y is given by . Positivity of V locally implies a Lipschitz bound on f; moreover, if intersects transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer the norms of and are controlled by these bounds, , and the smallness of . We give examples showing regularity extends from $X to part of , but not from Y to . We also show that when s remains nested for all (f,g), the problem in reduces to a supermodular problem in . © 2017 Wiley Periodicals, Inc.     

On the Lorentz-Marcinkiewicz Operator Ideal

This article deals with the LORENTZ-MARCINKIEWICZ operator ideal ?? generated by an additive s-function and the LORENTZ-MARCINKIEWICZ sequence space λ^q(φ). We give eigenvalue distributions for operators belonging to ?? (E, E) and we show the interpolation properties of ??-ideals. Furthermore, we study certain SCHAUDER bases in ?? (H, K), H and K Hilbert spaces.     

m‐ary Search trees when m ≥ 27: A strong asymptotics for the space requirements

It is known that the joint distribution of the number of nodes of each type of an m‐ary search tree is asymptotically multivariate normal when m ≤ 26. When m ≥ 27, we show the following strong asymptotics of the random vector X_n = ^t(X, … , X), where X denotes the number of nodes containing i ? 1 keys after having introduced n ? 1 keys in the tree: There exist (nonrandom) vectors X, C, and S and random variables ρ and φ such that (X_n ? nX)/n ? ρ(C cos(τ₂log n + φ) + S sin(τ₂log n + φ)) →_n→∞ 0 almost surely and in L²; σ₂ and τ₂ denote the real and imaginary parts of one of the eigenvalues of the transition matrix, having the second greatest real part. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Superconvergence of a 3D finite element method for stationary Stokes and Navier‐Stokes problems

For the Poisson equation on rectangular and brick meshes it is well known that the piecewise linear conforming finite element solution approximates the interpolant to a higher order than the solution itself. In this article, this type of supercloseness property is established for a special interpolant of the Q₂ ? P element applied to the 3D stationary Stokes and Navier‐Stokes problem, respectively. Moreover, applying a Q₃ ? P postprocessing technique, we can also state a superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself. Finally, we show that inhomogeneous boundary values can be approximated by the Lagrange Q₂‐interpolation without influencing the superconvergence property. Numerical experiments verify the predicted convergence rates. Moreover, a cost‐benefit analysis between the two third‐order methods, the post‐processed Q₂ ? P discretization, and the Q₃ ? P discretization is carried out. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

On the well‐posedness of the Cauchy problem for an MHD system in Besov spaces

This paper is devoted to the study of the Cauchy problem of incompressible magneto‐hydrodynamics system in the framework of Besov spaces. In the case of spatial dimension n?3, we establish the global well‐posedness of the Cauchy problem of an incompressible magneto‐hydrodynamics system for small data and the local one for large data in the Besov space ? (?ⁿ), 1?p<∞ and 1?r?∞. Meanwhile, we also prove the weak–strong uniqueness of solutions with data in ? (?ⁿ)∩L²(?ⁿ) for n/2p+2/r>1. In the case of n=2, we establish the global well‐posedness of solutions for large initial data in homogeneous Besov space ? (?²) for 2<p<∞ and 1?r<∞. Copyright © 2008 John Wiley & Sons, Ltd.     

Über die Radon-Transformation kreissymmetrischer Funktionen und ihre Beziehung zur Sommerfeldschen Theorie der Hankelfunktionen

For the Radon transform of functions with circular symmetry an inversion formula is proved in a new and elementary way. The inversion formula combined with Fourier theory is applied to Sommer-feld's integral for H, yielding a representation of products which generalizes Nicholson's integral for |H| ².     

Application of the τ‐function theory of Painlevé equations to random matrices: PV,PIII, the LUE,JUE, and CUE

With 〈·〉 denoting an average with respect to the eigenvalue PDF for the Laguerre unitary ensemble, the object of our study is for I = (0, s) and I = (s, ∞), where χ = 1 for λ_l ∈ I and χ = 0 otherwise. Using Okamoto's development of the theory of the Painlevé V equation, it is shown that ?_N(I; a, μ) is a τ‐function associated with the Hamiltonian therein, and so can be characterized as the solution of a certain second‐order second‐degree differential equation, or in terms of the solution of certain difference equations. The cases μ = 0 and μ = 2 are of particular interest, because they correspond to the cumulative distribution and density function, respectively, for the smallest and largest eigenvalue. In the case I = (s, ∞), ?_N(I; a, μ) is simply related to an average in the Jacobi unitary ensemble, and this in turn is simply related to certain averages over the orthogonal group, the unitary symplectic group, and the circular unitary ensemble. The latter integrals are of interest for their combinatorial content. Also considered are the hard‐edge and soft‐edge scaled limits of ?_N(I; a, μ). In particular, in the hard‐edge scaled limit it is shown that the limiting quantity ?^hard((O, s); a, μ) can be evaluated as a τ‐function associated with the Hamiltonian in Okamoto's theory of the Painlevé III equation. © 2002 Wiley Periodicals, Inc.     

About smoothness of solutions of the heat equations in closed,smooth space‐time domains

We consider the probabilistic solutions of the heat equation u = u + f in D, where D is a bounded domain in ?² = {(x¹, x²)} of class C^2k. We give sufficient conditions for u to have k^th‐order continuous derivatives with respect to (x¹, x²) in D? for integers k ≥ 2. The equation is supplemented with C^2k boundary data, and we assume that f ? C^2(k?1). We also prove that our conditions are sharp by examples in the border cases. © 2005 Wiley Periodicals, Inc.     

A new construction of perfect nonlinear functions using Galois rings

For a prime p, we give a construction of perfect nonlinear functions from ? to ? when either of the following conditions holds: (1) n≥p; (2) n<p, and n is a composite number or is the sum of positive composite numbers. It follows that when n≥12, there is a perfect nonlinear function from ? to ? for any prime p. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 229‐239, 2009     

On the global well‐posedness of discrete Boltzmann systems with chemical reaction

We study an initial‐boundary value problem in one‐space dimension for the discrete Boltzmann equation extended to a diatomic gas undergoing both elastic multiple collisions and chemical reactions. By integration of conservation equations, we prove a global existence result in the half‐space for small initial data N₀∈??∩L¹. Copyright © 2005 John Wiley & Sons, Ltd.     

Blow‐up estimates for a quasi‐linear reaction–diffusion system

In this paper, some sufficient conditions under which the quasilinear elliptic system ‐div(∣?_u∣^p‐2?_u) = uv, ‐div(∣?_u∣^q‐2?_u) = uv in ?^N(N≥3) has no radially symmetric positive solution is derived. Then by using this non‐existence result, blow‐up estimates for a class of quasilinear reaction–diffusion systems u_t = div (∣?_u∣^p‐2?_u)+uv,vt = div(∣?_v∣^q‐2?_v) +uv with the homogeneous Dirichlet boundary value conditions are obtained. Copyright © 2003 John Wiley & Sons, Ltd.