Authors:Hua Gui Duan; Hui Liu; Yi Ming Long; Wei Wang Pages: 1 - 18 Abstract: Abstract In this paper, we prove that for every index perfect non-degenerate compact star-shaped hypersurface Σ ⊂ R2n , there exist at least n non-hyperbolic closed characteristics with even Maslovtype indices on Σ when n is even. When n is odd, there exist at least n closed characteristics with odd Maslov-type indices on Σ and at least (n−1) of them are non-hyperbolic. Here we call a compact star-shaped hypersurface Σ ⊂ R2n index perfect if it carries only finitely many geometrically distinct prime closed characteristics, and every prime closed characteristic (τ, y) on Σ possesses positive mean index and whose Maslov-type index i(y,m) of its m-th iterate satisfies i(y,m) ≠ −1 when n is even, and i(y,m) ∉ {−2,−1, 0} when n is odd for all m ∈ N. PubDate: 2018-01-01 DOI: 10.1007/s10114-016-6019-9 Issue No:Vol. 34, No. 1 (2018)

Authors:Junhwa Choi; John Coates Pages: 19 - 28 Abstract: Abstract The field \(K = \mathbb{Q}\left( {\sqrt { - 7} } \right)\) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X 0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X 0(49) by the quadratic extension \(KK(\sqrt M )/K\) , where M is any square free element of O with M ≡ 1 mod 4 and (M,7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F ∞ = K(E p∞), where E p∞ denotes the group of p∞-division points on E. Moreover, writing B for the twist of X 0(49) by \(K(\sqrt[4]{{ - 7}})/K\) , our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper. PubDate: 2018-01-01 DOI: 10.1007/s10114-017-6414-x Issue No:Vol. 34, No. 1 (2018)

Authors:Dong Dong; Xiao Chun Li Pages: 29 - 41 Abstract: Abstract We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range \(D = \left\{ {\left( {{p_1},{p_2},{p_3}} \right) \in {\mathbb{R}^3}:1 < {p_1},{p_2} < \infty ,\frac{1}{{{p_1}}} + \frac{1}{{{p_2}}} < \frac{3}{2},1 < {p_3} < \infty } \right\}\) . PubDate: 2018-01-01 DOI: 10.1007/s10114-017-6415-9 Issue No:Vol. 34, No. 1 (2018)

Authors:Richard L. Wheeden Pages: 42 - 62 Abstract: Abstract Let Q(x) be a nonnegative definite, symmetric matrix such that \(\sqrt {Q\left( x \right)} \) is Lipschitz continuous. Given a real-valued function b(x) and a weak solution u(x) of div(Q∇u) = b, we find sufficient conditions in order that \(\sqrt Q \nabla u\) has some first order smoothness. Specifically, if Ω is a bounded open set in R n , we study when the components of \(\sqrt Q \nabla u\) belong to the first order Sobolev space W Q 1,2 (Ω) defined by Sawyer and Wheeden. Alternately, we study when each of n first order Lipschitz vector field derivatives X i u has some first order smoothness if u is a weak solution in Ω of Σ i=1 n X i ′ X i u + b = 0. We do not assume that {X i } is a Hörmander collection of vector fields in Ω. The results signal ones for more general equations. PubDate: 2018-01-01 DOI: 10.1007/s10114-016-6407-1 Issue No:Vol. 34, No. 1 (2018)

Authors:Yi Zhou; Yi Zhu Pages: 63 - 78 Abstract: Abstract In this paper, we derive the global existence of smooth solutions of the 3D incompressible Euler equations with damping for a class of large initial data, whose Sobolev norms H s can be arbitrarily large for any s ≥ 0. The approach is through studying the quantity representing the difference between the vorticity and velocity. And also, we construct a family of large solutions for MHD equations with damping. PubDate: 2018-01-01 DOI: 10.1007/s10114-016-6271-z Issue No:Vol. 34, No. 1 (2018)

Authors:Yonatan Gutman; Wen Huang; Song Shao; Xiang Dong Ye Pages: 79 - 90 Abstract: Abstract The family of pairwise independently determined (PID) systems, i.e., those for which the independent joining is the only self joining with independent 2-marginals, is a class of systems for which the long standing open question by Rokhlin, of whether mixing implies mixing of all orders, has a positive answer. We show that in the class of weakly mixing PID one finds a positive answer for another long-standing open problem, whether the multiple ergodic averages $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {{f_1}} \left( {{T^n}x} \right)...{f_d}\left( {{T^{dn}}x} \right),N \to \infty $$ almost surely converge. PubDate: 2018-01-01 DOI: 10.1007/s10114-017-6366-1 Issue No:Vol. 34, No. 1 (2018)

Authors:Bo Ju Jiang; Xue Zhi Zhao Pages: 91 - 102 Abstract: Abstract We give a brief survey of some developments in Nielsen fixed point theory. After a look at early history and a digress to various generalizations, we confine ourselves to several topics on fixed points of self-maps on manifolds and polyhedra. Special attention is paid to connections with geometric group theory and dynamics, as well as some formal approaches. PubDate: 2018-01-01 DOI: 10.1007/s10114-017-6503-x Issue No:Vol. 34, No. 1 (2018)

Authors:János Kollár; Johannes Nicaise; Chen Yang Xu Pages: 103 - 113 Abstract: Abstract Given a family of Calabi–Yau varieties over the punctured disc or over the field of Laurent series, we show that, after a finite base change, the family can be extended across the origin while keeping the canonical class trivial. More generally, we prove similar extension results for families whose log-canonical class is semi-ample. We use these to show that the Berkovich and essential skeleta agree for smooth varieties over ℂ((t)) with semi-ample canonical class. PubDate: 2018-01-01 DOI: 10.1007/s10114-017-7048-8 Issue No:Vol. 34, No. 1 (2018)

Authors:Rui Gao; Wei Xiao Shen Pages: 114 - 138 Abstract: Abstract We consider a class of generalized Fibonacci unimodal maps for which the central return times {s n } satisfy that s n = s n−1 + κs n−2 for some κ ≥ 1. We show that such a unimodal map admits a unique absolutely continuous invariant probability with exactly stretched exponential decay of correlations if its critical order lies in (1, κ + 1). PubDate: 2018-01-01 DOI: 10.1007/s10114-017-6438-2 Issue No:Vol. 34, No. 1 (2018)

Authors:Xin Jun Gao Abstract: Abstract We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg–de Vries equation in Sobolev spaces H s (R) for s > \( - \frac{3}{{22}}\) . The main approach is the “I-method” together with the multilinear multiplier analysis. PubDate: 2018-01-12 DOI: 10.1007/s10114-018-7241-4

Authors:Hong Liang Feng; Hua Wang; Xiao Hua Yao Abstract: Abstract Based on the endpoint Strichartz estimates for the fourth order Schrödinger equation with potentials for n ≥ 5 by [Feng, H., Soffer, A., Yao, X.: Decay estimates and Strichartz estimates of the fourth-order Schrödinger operator. J. Funct. Anal., 274, 605–658 (2018)], in this paper, the authors further derive Strichartz type estimates with gain of derivatives similar to the one in [Pausader, B.: The cubic fourth-order Schrödinger equation. J. Funct. Anal., 256, 2473–2517 (2009)]. As their applications, we combine the classical Morawetz estimate and the interaction Morawetz estimate to establish scattering theory in the energy space for the defocusing fourth order NLS with potentials and pure power nonlinearity \(1 + \frac{8}{n} < p < 1 + \frac{8}{{n - 4}}\) in dimensions n ≥ 7. PubDate: 2018-01-12 DOI: 10.1007/s10114-018-7343-z

Authors:Jin Gang Xiong Abstract: Abstract We study a prescribing functions problem of a conformally invariant integral equation involving Poisson kernel on the unit ball. This integral equation is not the dual of any standard type of PDE. As in Nirenberg problem, there exists a Kazdan–Warner type obstruction to existence of solutions. We prove existence in the antipodal symmetry functions class. PubDate: 2018-01-12 DOI: 10.1007/s10114-018-7309-1

Authors:Yuan Yuan Nie; Chun Peng Wang Abstract: Abstract This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C 1/2 Hölder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary. PubDate: 2017-12-28 DOI: 10.1007/s10114-017-7341-6

Authors:Hao Jia; Vladimír Šverák Abstract: Abstract We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder. PubDate: 2017-12-28 DOI: 10.1007/s10114-017-7397-3

Authors:Wei Hua Wang; Gang Wu Abstract: Abstract In this paper, we establish the global well-posedness of the generalized rotating magnetohydrodynamics equations if the initial data are in X 1−2α defined by \({x^{1 - 2\alpha }} = \left\{ {u \in D'\left( {{R^3}} \right):{{\int_{{R^3}} {\left \xi \right } }^{1 - 2\alpha }}\left {\hat u\left( \xi \right)} \right d\xi < + \infty } \right\}\) . In addition, we also give Gevrey class regularity of the solution. PubDate: 2017-12-20 DOI: 10.1007/s10114-017-7276-y

Authors:Jaume Llibre; Clàudia Valls Abstract: Abstract We study the Hindmarsh–Rose burster which can be described by the differential system $$\dot x = y - {x^3} + b{x^2} + I - z,\dot y = 1 - 5{x^2} - y,\dot z = \mu \left( {s\left( {x - {x_0}} \right) - z} \right),$$ where b, I, μ, s, x 0 are parameters. We characterize all its invariant algebraic surfaces and all its exponential factors for all values of the parameters. We also characterize its Darboux integrability in function of the parameters. These characterizations allow to study the global dynamics of the system when such invariant algebraic surfaces exist. PubDate: 2017-12-20 DOI: 10.1007/s10114-017-5661-1

Authors:Meng Yun Liu; Cheng Bo Wang Abstract: Abstract In this paper, we prove the global existence for some 4-D quasilinear wave equations with small, radial data in H 3 × H 2. The main idea is to exploit local energy estimates with variable coefficients, together with the trace estimates. PubDate: 2017-12-20 DOI: 10.1007/s10114-017-7138-7

Authors:Aung Zaw Myint; Li Li; Ming Xin Wang Abstract: Abstract This paper deals with one kind of Belousov–Zhabotinskii reaction model. Linear stability is discussed for the spatially homogeneous problem firstly. Then we focus on the stationary problem with diffusion. Non-existence and existence of non-constant positive solutions are obtained by using implicit function theorem and Leray–Schauder degree theory, respectively. PubDate: 2017-12-20 DOI: 10.1007/s10114-017-7295-8

Authors:Yu Kang Chen; Zhen Lei; Chang Hua Wei Abstract: Abstract Caffarelli and Silvestre [Comm. Part. Diff. Eqs., 32, 1245–1260 (2007)] characterized the fractional Laplacian (−Δ) s as an operator maps Dirichlet boundary condition to Neumann condition via the harmonic extension problem to the upper half space for 0 < s < 1. In this paper, we extend this result to all s > 0. We also give a new proof to the dissipative a priori estimate of quasi-geostrophic equations in the framework of L p norm using the Caffarelli–Silvestre’s extension technique. PubDate: 2017-12-20 DOI: 10.1007/s10114-017-7325-6

Authors:Chao Lu; Jing Lu Abstract: Abstract In this paper, we study the unconditional uniqueness of solution for the Cauchy problem of \(\dot H^{s_c } (0 \leqslant s_c < 2)\) critical nonlinear fourth-order Schrödinger equations i∂ t u+Δ2 u−ϵu = λ u α u. By employing paraproduct estimates and Strichartz estimates, we prove that unconditional uniqueness of solution holds in \(C_t (I;\dot H^{s_c } (\mathbb{R}^d ))\) for d ≥ 11 and \(\min \left\{ {1^ - ,\tfrac{8} {{d - 4}}} \right\} \geqslant \alpha > \frac{{ - (d - 4) + \sqrt {(d - 4)^2 + 64} }} {4}\) . PubDate: 2017-12-20 DOI: 10.1007/s10114-017-7354-1