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In this paper, we prove that the 1D Cauchy problem of the compressible Navier–Stokes equations admits a unique global classical solution (ρ,u) if the viscosity μ(ρ)=1+ρβ with β?0. The initial data can be arbitrarily large and may contain vacuum. Some new weighted estimates of the density and velocity are obtained when deriving higher order estimates of the solution. 相似文献
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Jean-Pierre Kahane 《Comptes Rendus Mathematique》2014,352(5):383-385
For almost all x>1, (xn)(n=1,2,…) is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (bn) in [0,1[ and ε>0, the x -set such that |xn−bn|<ε modulo 1 for n large enough has dimension 1. However, its intersection with an interval [1,X] has a dimension <1, depending on ε and X. Some results are given and a question is proposed. 相似文献
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Michał Kowalczyk Yong Liu Frank Pacard 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2012
We are interested in entire solutions of the Allen–Cahn equation Δu−F′(u)=0 which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4-ended solutions . The main result of our paper states that, for any θ∈(0,π/2), there exists a 4-ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ , π−θ, π+θ and 2π−θ with the x-axis. This paper is part of a program whose aim is to classify all 2k -ended solutions of the Allen–Cahn equation in dimension 2, for k?2. 相似文献
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We consider the semilinear elliptic equation Δu+K(|x|)up=0 in RN for N>2 and p>1, and study separation phenomena of positive radial solutions. With respect to intersection and separation, we establish a classification of the solution structures, and investigate the structures of intersection, partial separation and separation. As a consequence, we obtain the existence of positive solutions with slow decay when the oscillation of the function r−?K(r) with ?>−2 around a positive constant is small near r=∞ and p is sufficiently large. Moreover, if the assumptions hold in the whole space, the equation has the structure of separation and possesses a singular solution as the upper limit of regular solutions. We also reveal that the equation changes its nature drastically across a critical exponent pc which is determined by N and the order of the behavior of K(r) as r=|x|→0 and ∞. In order to understand how subtle the structure is on K at p=pc, we explain the criticality in a similar way as done by Ding and Ni (1985) [6] for the critical Sobolev exponent p=(N+2)/(N−2). 相似文献
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Let K be a closed convex subset of a q-uniformly smooth separable Banach space, T:K→K a strictly pseudocontractive mapping, and f:K→K an L-Lispschitzian strongly pseudocontractive mapping. For any t∈(0,1), let xt be the unique fixed point of tf+(1-t)T. We prove that if T has a fixed point, then {xt} converges to a fixed point of T as t approaches to 0. 相似文献
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The dimension of a point x in Euclidean space (meaning the constructive Hausdorff dimension of the singleton set {x}) is the algorithmic information density of x . Roughly speaking, this is the least real number dim(x) such that r×dim(x) bits suffice to specify x on a general-purpose computer with arbitrarily high precision 2−r. The dimension spectrum of a set X in Euclidean space is the subset of [0,n] consisting of the dimensions of all points in X. 相似文献
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The period annuli of the planar vector field x′=−yF(x,y), y′=xF(x,y), where the set {F(x,y)=0} consists of k different isolated points, is defined by k+1 concentric annuli. In this paper we perturb it with polynomials of degree n and we study how many limit cycles bifurcate, up to a first order analysis, from all the period annuli simultaneously in terms of k and n . Additionally, we prove that the associated Abelian integral is piecewise rational and, when k=1, the provided upper bound is reached. Finally, the case k=2 is also treated. 相似文献
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We prove the global existence of the small solutions to the Cauchy problem for quasilinear wave equations satisfying the null condition on (R3,g), where the metric g is a small perturbation of the flat metric and approaches the Euclidean metric like (1+|x|2)−ρ/2 with ρ>1. Global and almost global existence for systems without the null condition are also discussed for certain small time-dependent perturbations of the flat metric in Appendix?A. 相似文献
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We consider the Cauchy problem for the generalized Ostrovsky equation where f(u)=|u|ρ−1u if ρ is not an integer and f(u)=uρ if ρ is an integer. We obtain the L∞ time decay estimates and the large time asymptotics of small solutions under suitable conditions on the initial data and the order of the nonlinearity. 相似文献
utx=u+(f(u))xx,
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In this paper we study the equation −Δu+ρ−(α+2)h(ραu)=0 in a smooth bounded domain Ω where ρ(x)=dist(x,∂Ω), α>0 and h is a nondecreasing function which satisfies Keller–Osserman condition. We introduce a condition on h which implies that the equation is subcritical, i.e., the corresponding boundary value problem is well posed with respect to data given by finite measures. Under additional assumptions on h we show that this condition is necessary as well as sufficient. We also discuss b.v. problems with data given by positive unbounded measures. Our results extend results of [13] treating equations of the form −Δu+ρβuq=0 with q>1, β>−2. 相似文献
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New representation and factorizations of the higher-order ultraspherical-type differential equations
The paper deals with the class of linear differential equations of any even order 2α+4, α∈N0, which are associated with the so-called ultraspherical-type polynomials. These polynomials form an orthogonal system on the interval [−1,1] with respect to the ultraspherical weight function (1−x2)α and additional point masses of equal size at the two endpoints. The differential equations of “ultraspherical-type” were developed by R. Koekoek in 1994 by utilizing special function methods. In the present paper, a new and completely elementary representation of these higher-order differential equations is presented. This result is used to deduce the orthogonality relation of the ultraspherical-type polynomials directly from the differential equation property. Moreover, we introduce two types of factorizations of the corresponding differential operators of order 2α+4 into a product of α+2 linear second-order operators. 相似文献
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Let G be a simple connected graph of order n with degree sequence d1,d2,…,dn in non-increasing order. The signless Laplacian spectral radius ρ(Q(G)) of G is the largest eigenvalue of its signless Laplacian matrix Q(G). In this paper, we give a sharp upper bound on the signless Laplacian spectral radius ρ(Q(G)) in terms of di, which improves and generalizes some known results. 相似文献