Authors:Huai-Liang Chang; Jun Li; Wei-Ping Li; Chiu-Chu Melissa Liu Pages: 869 - 882 Abstract: The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables. It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants. Then it brie y discusses the master space technique and its application to the set-up of the MSP moduli. Some key results in MSP theory are explained and some examples are provided. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1101-x Issue No:Vol. 38, No. 4 (2017)

Authors:Jinwon Choi; Young-Hoon Kiem Pages: 883 - 900 Abstract: Landau-Ginzburg/Calabi-Yau correspondence claims the equivalence between the Gromov-Witten theory and the Fan-Jarvis-Ruan-Witten theory. The authors survey recently developed wall-crossing approach to the Landau-Ginzburg/Calabi-Yau correspondence for a quintic threefold. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1102-9 Issue No:Vol. 38, No. 4 (2017)

Authors:Richard Eager; Kentaro Hori; Johanna Knapp; Mauricio Romo Pages: 901 - 912 Abstract: The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1103-8 Issue No:Vol. 38, No. 4 (2017)

Authors:Huijun Fan; Tyler Jarvis; Yongbin Ruan Pages: 913 - 936 Abstract: This is a survey article for the mathematical theory of Witten’s Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper the authors focus on the description of the moduli. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1104-7 Issue No:Vol. 38, No. 4 (2017)

Authors:Mauricio Romo Pages: 937 - 962 Abstract: The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. The author focuses on the case of G = SL(N, C) and M being a knot complement: M = S 3 \ K. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G = SL(N, C). He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1105-6 Issue No:Vol. 38, No. 4 (2017)

Authors:Yongbin Ruan Pages: 963 - 984 Abstract: The gauged linear sigma model (GLSM for short) is a 2d quantum field theory introduced by Witten twenty years ago. Since then, it has been investigated extensively in physics by Hori and others. Recently, an algebro-geometric theory (for both abelian and nonabelian GLSMs) was developed by the author and his collaborators so that he can start to rigorously compute its invariants and check against physical predications. The abelian GLSM was relatively better understood and is the focus of current mathematical investigation. In this article, the author would like to look over the horizon and consider the nonabelian GLSM. The nonabelian case possesses some new features unavailable to the abelian GLSM. To aid the future mathematical development, the author surveys some of the key problems inspired by physics in the nonabelian GLSM. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1106-5 Issue No:Vol. 38, No. 4 (2017)

Authors:Aydin Gezer; Abdullah Magden Pages: 985 - 998 Abstract: Let (M, g) be an n-dimensional Riemannian manifold and T 2 M be its second-order tangent bundle equipped with a lift metric \(\tilde g\) . In this paper, first, the authors construct some Riemannian almost product structures on (T 2 M, \(\tilde g\) ) and present some results concerning these structures. Then, they investigate the curvature properties of (T 2 M, \(\tilde g\) ). Finally, they study the properties of two metric connections with nonvanishing torsion on (T 2 M, \(\tilde g\) ): The H-lift of the Levi-Civita connection of g to T 2 M, and the product conjugate connection defined by the Levi-Civita connection of \(\tilde g\) and an almost product structure. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1107-4 Issue No:Vol. 38, No. 4 (2017)

Authors:Yu Lu; Shenglin Zhu Pages: 999 - 1018 Abstract: In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/A H a right H*-Galois extension. The authors prove that, if A H is a separable k-algebra, then for any right coideal subalgebra B of H, the B-invariants A B = {a ∈ A b · a = ε(b)a, ∀b ∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing A H as in the classical case. The results are applied to the case H = (kG)* for a finite group G to get a Galois 1-1 correspondence. PubDate: 2017-07-01 DOI: 10.1007/s11401-017-1108-3 Issue No:Vol. 38, No. 4 (2017)

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: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: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)