Motivated by the work of Birman about the relationship between mapping class groups and braid groups, the authors discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface M with a free and proper group action in this paper. Their construction is based on the exact sequence given by the fibration ${\cal F}_0^GM \to F\left( {M/G,n} \right)$. The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is ${\mathbb{T}^2}$.

Let u(t, x) be the solution to the Cauchy problem of a scalar conservation law in one space dimension. It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations. It is impossible that the solutions belong to, for example, H ¹ because by Sobolev embedding theorem H ¹ functions are Hölder continuous. However, the author notes that from any point (t, x), he can draw a generalized characteristic downward which meets the initial axis at y = α(t, x). If he regards u as a function of (t, y), it indeed belongs to H ¹ as a function of y if the initial data belongs to H ¹. He may call this generalized persistence (of high regularity) of the entropy weak solutions. The main purpose of this paper is to prove some kinds of generalized persistence (of high regularity) for the scalar and 2 × 2 Temple system of hyperbolic conservation laws in one space dimension.

In this paper, the authors use Glimm scheme to study the global existence of BV solutions to Cauchy problem of the pressure-gradient system with large initial data. To this end, some important properties of the shock curves of the pressure-gradient system in the Riemann invariant coordinate system and verify that the shock curves satisfy Diperna’s conditions (see [Diperna, R. J., Existence in the large for quasilinear hyperbolic conservation laws, Arch. Ration. Mech. Anal., 52(3), 1973, 244–257]) are studied. Then they construct the approximate solution sequence through Glimm scheme. By establishing accurate local interaction estimates, they prove the boundedness of the approximate solution sequence and its total variation.

In this paper, the authors consider the mean field game with a common noise and allow the state coefficients to vary with the conditional distribution in a nonlinear way. They assume that the cost function satisfies a convexity and a weak monotonicity property. They use the sufficient Pontryagin principle for optimality to transform the mean field control problem into existence and uniqueness of solution of conditional distribution dependent forward-backward stochastic differential equation (FBSDE for short). They prove the existence and uniqueness of solution of the conditional distribution dependent FBSDE when the dependence of the state on the conditional distribution is sufficiently small, or when the convexity parameter of the running cost on the control is sufficiently large. Two different methods are developed. The first method is based on a continuation of the coefficients, which is developed for FBSDE by [Hu, Y. and Peng, S., Solution of forward-backward stochastic differential equations, Probab. Theory Rel., 103(2), 1995, 273–283]. They apply the method to conditional distribution dependent FBSDE. The second method is to show the existence result on a small time interval by Banach fixed point theorem and then extend the local solution to the whole time interval.

In this paper, the authors consider the following singular Kirchhoff-Schrödinger problem$M\left( {\int_{{\mathbb{R}^N}} {{{\left| {\nabla u} \right|}^N} + V\left( x \right){{\left| u \right|}^N}{\rm{d}}x} } \right)\left( { - {\Delta _N}u + V\left( x \right){{\left| u \right|}^{N - 2}}u} \right) = {{f\left( {x,u} \right)} \over {{{\left| x \right|}^\eta }}}\,\,\,\,\,{\rm{in}}\,\,{\mathbb{R}^N},\,\,\,\,\,\,\,\left( {{P_\eta }} \right)$

where 0 < η < N, M is a Kirchhoff-type function and V(x) is a continuous function with positive lower bound, f(x, t) has a critical exponential growth behavior at infinity. Combining variational techniques with some estimates, they get the existence of ground state solution for (P _η)- Moreover, they also get the same result without the A-R condition.

In this paper, by using Seshadri constants for subschemes, the author establishes a second main theorem of Nevanlinna theory for holomorphic curves intersecting closed subschemes in (weak) subgeneral position. As an application of his second main theorem, he obtain a Brody hyperbolicity result for the complement of nef effective divisors. He also give the corresponding Schmidt’s subspace theorem and arithmetic hyperbolicity result in Diophantine approximation.

The authors study the continuity estimate of the solutions of almost harmonic maps with the perturbation term f in a critical integrability class (Zygmund class) ${L^{{n \over 2}}}$, log ^q L, n is the dimension with n ≥ 3. They prove that when $q > {n \over 2}$ the solution must be continuous and they can get continuity modulus estimates. As a byproduct of their method, they also study boundary continuity for the almost harmonic maps in high dimension.

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold M ⁿ × ℝ, where M ⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and ℝ is the Euclidean 1-space.

In this paper, the authors first introduce the tree-indexed Markov chains in random environment, which takes values on a general state space. Then, they prove the existence of this stochastic process, and develop a class of its equivalent forms. Based on this property, some strong limit theorems including conditional entropy density are studied for the tree-indexed Markov chains in random environment.