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 Chinese Annals of Mathematics, Series B   [SJR: 0.445]   [H-I: 25]   [0 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 0252-9599 - ISSN (Online) 1860-6261    Published by Springer-Verlag  [2345 journals]
• Exact boundary controllability on a tree-like network of nonlinear planar
Timoshenko beams
• Authors: Qilong Gu; Günter Leugering; Tatsien Li
Pages: 711 - 740
Abstract: This paper concerns a system of equations describing the vibrations of a planar network of nonlinear Timoshenko beams. The authors derive the equations and appropriate nodal conditions, determine equilibrium solutions and, using the methods of quasilinear hyperbolic systems, prove that for tree-like networks the natural initial-boundary value problem admits semi-global classical solutions in the sense of Li [Li, T. T., Controllability and Observability for Quasilinear Hyperbolic Systems, AIMS Ser. Appl. Math., vol 3, American Institute of Mathematical Sciences and Higher Education Press, 2010] existing in a neighborhood of the equilibrium solution. The authors then prove the local exact controllability of such networks near such equilibrium configurations in a certain specified time interval depending on the speed of propagation in the individual beams.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1092-7
Issue No: Vol. 38, No. 3 (2017)

• Blowup criteria for full compressible Navier-Stokes equations with vacuum
state
• Authors: Yongfu Wang; Shan Li
Pages: 741 - 758
Abstract: This paper deals with the global strong solution to the three-dimensional (3D) full compressible Navier-Stokes systems with vacuum. The authors provide a sufficient condition which requires that the Sobolev norm of the temperature and some norm of the divergence of the velocity are bounded, for the global regularity of strong solution to the 3D compressible Navier-Stokes equations. This result indicates that the divergence of velocity fields plays a dominant role in the blowup mechanism for the full compressible Navier-Stokes equations in three dimensions.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1093-6
Issue No: Vol. 38, No. 3 (2017)

• Quasi-periodic solutions for the derivative nonlinear Schrödinger
equation with finitely differentiable nonlinearities
• Authors: Meina Gao; Kangkang Zhang
Pages: 759 - 786
Abstract: The authors are concerned with a class of derivative nonlinear Schrödinger equation $$i{u_t} + {u_{xx}} + i \epsilon f\left( {u,\bar u,\omega t} \right){u_x} = 0,\left( {t,x} \right) \in \mathbb{R} \times \left[ {0,\pi } \right],$$ subject to Dirichlet boundary condition, where the nonlinearity f(z 1, z 2, ϕ) is merely finitely differentiable with respect to all variables rather than analytic and quasi-periodically forced in time. By developing a smoothing and approximation theory, the existence of many quasi-periodic solutions of the above equation is proved.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1094-5
Issue No: Vol. 38, No. 3 (2017)

• Global well-posedness of incompressible Navier-Stokes equations with two
slow variables
• Authors: Weimin Peng; Yi Zhou
Pages: 787 - 794
Abstract: In this paper, the global well-posedness of the three-dimensional incompressible Navier-Stokes equations with a linear damping for a class of large initial data slowly varying in two directions are proved by means of a simpler approach.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1095-4
Issue No: Vol. 38, No. 3 (2017)

• Weighted compact commutator of bilinear Fourier multiplier operator
• Authors: Guoen Hu
Pages: 795 - 814
Abstract: Let T σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that $$\mathop {\sup }\limits_{k \in \mathbb{Z}} {\left\ {{\sigma _k}} \right\ _{{W^s}\left( {{\mathbb{R}^{2n}}} \right)}} < \infty$$ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T σ and CMO(ℝ n ) functions is a compact operator from $${L^{{p_1}}}\left( {{\mathbb{R}^n},{\omega _1}} \right) \times {L^{{p_2}}}\left( {{\mathbb{R}^n},{\omega _2}} \right)$$ to $${L^p}\left( {{\mathbb{R}^n},{\nu _{\vec \omega }}} \right)$$ for appropriate indices p 1, p 2, p ∈ (1,∞) with $$\frac{1}{p} = \frac{1}{{{p_1}}} + \frac{1}{{{p_2}}}$$ and weights ω 1,ω 2 such that $$\vec \omega = \left( {{\omega _1},{\omega _2}} \right) \in {A_{\vec p/\vec t}}\left( {{\mathbb{R}^{2n}}} \right)$$ .
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1096-3
Issue No: Vol. 38, No. 3 (2017)

• Singularity of the extremal solution for supercritical biharmonic
equations with power-type nonlinearity
• Authors: Baishun Lai; Zhengxiang Yan; Yinghui Zhang
Pages: 815 - 826
Abstract: Let B ⊂ ℝ n be the unit ball centered at the origin. The authors consider the following biharmonic equation: $$\left\{ {\begin{array}{*{20}{c}} {{\Delta ^2}u = \lambda {{\left( {1 + u} \right)}^p}}&{in \mathbb{B},} \\ {u = \frac{{\partial u}}{{\partial \nu }} = 0}&{on\partial \mathbb{B},} \end{array}} \right.$$ where $$p > \frac{{n + 4}}{{n - 4}}$$ and v is the outward unit normal vector. It is well-known that there exists a λ* > 0 such that the biharmonic equation has a solution for λ ∈ (0, λ*) and has a unique weak solution u* with parameter λ = λ*, called the extremal solution. It is proved that u* is singular when n ≥ 13 for p large enough and satisfies $$u* \leqslant {r^{ - \frac{4}{{p - 1}}}} - 1$$ on the unit ball, which actually solve a part of the open problem left in [Dàvila, J., Flores, I., Guerra, I., Multiplicity of solutions for a fourth order equation with power-type nonlinearity, Math. Ann., 348(1), 2009, 143–193].
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1097-2
Issue No: Vol. 38, No. 3 (2017)

• Blow up for initial-boundary value problem of wave equation with a
nonlinear memory in 1-D
• Authors: Ning-An Lai; Jianli Liu; Jinglei Zhao
Pages: 827 - 838
Abstract: The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: $${u_{tt}} - {u_{xx}} = \frac{1}{{\Gamma \left( {1 - \gamma } \right)}}\int_0^t {{{\left( {t - s} \right)}^{ - \gamma }}{{\left {u\left( s \right)} \right }^p}ds} .$$ The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1098-1
Issue No: Vol. 38, No. 3 (2017)

• Fractional Sobolev-Poincaré inequalities in irregular domains
• Authors: Chang-Yu Guo
Pages: 839 - 856
Abstract: This paper is devoted to the study of fractional (q, p)-Sobolev-Poincaré in- equalities in irregular domains. In particular, the author establishes (essentially) sharp fractional (q, p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional (q, p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P., Sobolev-Poincaré implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1099-0
Issue No: Vol. 38, No. 3 (2017)

• Optimal transportation for generalized Lagrangian
• Authors: Ji Li; Jianlu Zhang
Pages: 857 - 868
Abstract: This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: $$c\left( {x,y} \right) = \mathop {\inf }\limits_{\begin{array}{*{20}{c}} {x\left( 0 \right) = x} \\ {x\left( 1 \right) = y} \\ {u \in U} \end{array}} \int_0^1 {L\left( {x\left( s \right),u\left( {x\left( s \right),s} \right),s} \right)ds} ,$$ where U is a control set, and x satisfies the ordinary equation $$\dot x\left( s \right) = f\left( {x\left( s \right),u\left( {x\left( s \right),s} \right)} \right).$$ It is proved that under the condition that the initial measure μ 0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: $$\left\{ {_{V\left( {0,x} \right) = {\phi _0}\left( x \right).}^{{V_t}\left( {t,x} \right) + \mathop {\sup }\limits_{u \in U} \left\langle {{V_x}\left( {t,x} \right),f\left( {x,u\left( {x\left( t \right),t} \right),t} \right) - L\left( {x\left( t \right),u\left( {x\left( t \right),t} \right),t} \right)} \right\rangle = 0,}} \right.$$
PubDate: 2017-05-01
DOI: 10.1007/s11401-017-1100-y
Issue No: Vol. 38, No. 3 (2017)

• New identities for Weak KAM theory
• Authors: Lawrence Craig Evans
Pages: 379 - 392
Abstract: This paper records for the Hamiltonian H = 1/2 p 2 + W(x) some old and new identities relevant for the PDE/variational approach to weak KAM theory.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1074-9
Issue No: Vol. 38, No. 2 (2017)

• Quantitative stability of the Brunn-Minkowski inequality for sets of equal
volume
• Authors: Alessio Figalli; David Jerison
Pages: 393 - 412
Abstract: The authors prove a quantitative stability result for the Brunn-Minkowski inequality on sets of equal volume: If A = B > 0 and A + B 1/n = (2+δ) A 1/n for some small δ, then, up to a translation, both A and B are close (in terms of δ) to a convex set K. Although this result was already proved by the authors in a previous paper, the present paper provides a more elementary proof that the authors believe has its own interest. Also, the result here provides a stronger estimate for the stability exponent than the previous result of the authors.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1075-8
Issue No: Vol. 38, No. 2 (2017)

• Affinely prime dynamical systems
• Authors: Hillel Furstenberg; Eli Glasner; Benjamin Weiss
Pages: 413 - 424
Abstract: This paper deals with representations of groups by “affine” automorphisms of compact, convex spaces, with special focus on “irreducible” representations: equivalently “minimal” actions. When the group in question is PSL(2, R), the authors exhibit a one-one correspondence between bounded harmonic functions on the upper half-plane and a certain class of irreducible representations. This analysis shows that, surprisingly, all these representations are equivalent. In fact, it is found that all irreducible affine representations of this group are equivalent. The key to this is a property called “linear Stone-Weierstrass” for group actions on compact spaces. If it holds for the “universal strongly proximal space” of the group (to be defined), then the induced action on the space of probability measures on this space is the unique irreducible affine representation of the group.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1076-7
Issue No: Vol. 38, No. 2 (2017)

• Exact boundary controllability for a coupled system of wave equations with
Neumann boundary controls
• Authors: Tatsien Li; Bopeng Rao
Pages: 473 - 488
Abstract: This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next, in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1078-5
Issue No: Vol. 38, No. 2 (2017)

• On the C 1 regularity of solutions to divergence form elliptic systems
with dini-continuous coefficients
• Authors: Yanyan Li
Pages: 489 - 496
Abstract: The author proves C 1 regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1079-4
Issue No: Vol. 38, No. 2 (2017)

• Bolzano’s theorems for holomorphic mappings
• Authors: Jean Mawhin
Pages: 563 - 578
Abstract: The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano’s ones for real functions on an interval is deduced in a very simple way from Cauchy’s theorem for holomorphic functions. A more complicated proof, using Cauchy’s argument principle, provides uniqueness of the zero, when the sign conditions on the boundary are strict. Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions. Extensions to holomorphic mappings from ℂ n to ℂ n are obtained using Brouwer degree.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1083-8
Issue No: Vol. 38, No. 2 (2017)

• Asymptotics and blow-up for mass critical nonlinear dispersive equations
• Authors: Frank Merle
Pages: 579 - 590
Abstract: The author considers mass critical nonlinear Schrödinger and Korteweg-de Vries equations. A review on results related to the blow-up of solution of these equations is given.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1084-7
Issue No: Vol. 38, No. 2 (2017)

• Singular solutions to conformal Hessian equations
• Authors: Nikolai Nadirashvili; Serge Vlăduţ
Pages: 591 - 600
Abstract: The authors show that for any ε ∈]0, 1[, there exists an analytic outside zero solution to a uniformly elliptic conformal Hessian equation in a ball B ⊂ ℝ5 which belongs to C 1,ε (B) \C 1,ε +(B).
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1085-6
Issue No: Vol. 38, No. 2 (2017)

• Convergence to a single wave in the Fisher-KPP equation
• Authors: James Nolen; Jean-Michel Roquejoffre; Lenya Ryzhik
Pages: 629 - 646
Abstract: The authors study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t−(3/2)log t+x ∞, the solution of the equation converges as t → +∞ to a translate of the traveling wave corresponding to the minimal speed c * = 2. The constant x ∞ depends on the initial condition u(0, x). The proof is elaborate, and based on probabilistic arguments. The purpose of this paper is to provide a simple proof based on PDE arguments.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1087-4
Issue No: Vol. 38, No. 2 (2017)

• A third derivative estimate for Monge-Ampere equations with conic
singularities
• Authors: Gang Tian
Pages: 687 - 694
Abstract: The author applies the arguments in his PKU Master degree thesis in 1988 to derive a third derivative estimate, and consequently, a C 2,α -estimate, for complex Monge-Ampere equations in the conic case. This C 2,α -estimate was used by Jeffres-Mazzeo-Rubinstein in their proof of the existence of Kähler-Einstein metrics with conic singularities.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1090-9
Issue No: Vol. 38, No. 2 (2017)

• CR geometry in 3-D
• Authors: Paul C. Yang
Pages: 695 - 710
Abstract: CR geometry studies the boundary of pseudo-convex manifolds. By concentrating on a choice of a contact form, the local geometry bears strong resemblence to conformal geometry. This paper deals with the role conformally invariant operators such as the Paneitz operator plays in the CR geometry in dimension three. While the sign of this operator is important in the embedding problem, the kernel of this operator is also closely connected with the stability of CR structures. The positivity of the CR-mass under the natural sign conditions of the Paneitz operator and the CR Yamabe operator is discussed. The CR positive mass theorem has a consequence for the existence of minimizer of the CR Yamabe problem. The pseudo-Einstein condition studied by Lee has a natural analogue in this dimension, and it is closely connected with the pluriharmonic functions. The author discusses the introduction of new conformally covariant operator P-prime and its associated Q-prime curvature and gives another natural way to find a canonical contact form among the class of pseudo-Einstein contact forms. Finally, an isoperimetric constant determined by the Q-prime curvature integral is discussed.
PubDate: 2017-03-01
DOI: 10.1007/s11401-017-1091-8
Issue No: Vol. 38, No. 2 (2017)

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