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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)