Convergence Rates for Discretized Monge–Ampère Equations and Quantitative Stability of Optimal Transport

Foundations of Computational Mathematics - In recent works—both experimental and theoretical—it has been shown how to use computational geometry to efficiently construct approximations...     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

On the persistence and blow up for the generalized two-component Dullin–Gottwald–Holm system

Considered herein is the persistence property of the solutions to the generalized two-component integrable Dullin–Gottwald–Holm system, which was derived from the Euler equation with nonzero constant vorticity in shallow water waves moving over a linear shear flow. Firstly, the persistence properties of the system are investigated in weighted $$L^p$$-spaces for a large class of moderate weights. Then, we establish the new local-in-space blow-up results simplifying and extending earlier blow-up criterion for this system.     

Homogenization Theory for a Replenishing Passive Scalar Field

Homogenization theory provides a rigorous framework for calculating the effective diffusivity of a decaying passive scalar field in a turbulent or complex flow. The authors extend this framework to the case where the passive scalar fluctuations are continuously replenished by a source （and/or sink）. The basic structure of the homogenized equations carries over, but in some cases the homogenized source can involve a non-trivial coupling of the velocity field and the source. The authors derive expressions for the homogenized source term for various multiscale source structures and interpret them physically.     

On the Motion Law of Fronts for Scalar Reaction-Diffusion Equations with Equal Depth Multiple-Well Potentials

Slow motion for scalar Allen-Cahn type equation is a well-known phenomenon, precise motion law for the dynamics of fronts having been established first using the socalled geometric approach inspired from central manifold theory (see the results of Carr and Pego in 1989). In this paper, the authors present an alternate approach to recover the motion law, and extend it to the case of multiple wells. This method is based on the localized energy identity, and is therefore, at least conceptually, simpler to implement. It also allows to handle collisions and rough initial data.     

A Nash–Moser approach for the Euler–Arnold equations

We study the local well-posedness in the smooth category for a class of Euler equations. A Nash–Moser approach is used to extend, for the case of an invertible elliptic pseudo-differential operator, some results obtained by Escher and Kolev, with the help of some geometric arguments.     

On the Well-Posedness of Stochastic Boussinesq Equations with Transport Noise

Journal of Nonlinear Science - The Boussinesq equations play a fundamental role in meteorology. Among other aspects, they aim to model the process of frontogenesis and describe large-scale...     

Initial Boundary Value Problems for Scalar and Vector Burgers Equations

In this article we study Burgers equation and vector Burgers equation with initial and boundary conditions. First we consider the Burgers equation in the quarter plane x >0, t >0 with Riemann type of initial and boundary conditions and use the Hopf–Cole transformation to linearize the problems and explicitly solve them. We study two limits, the small viscosity limit and the large time behavior of solutions. Next, we study the vector Burgers equation and solve the initial value problem for it when the initial data are gradient of a scalar function. We investigate the asymptotic behavior of this solution as time tends to infinity and generalize a result of Hopf to the vector case. Then we construct the exact N-wave solution as an asymptote of solution of an initial value problem extending the previous work of Sachdev et al. (1994). We also study the limit as viscosity parameter goes to 0.Finally, we get an explicit solution for a boundary value problem in a cylinder.     

非线性耦合标量场方程显式解析解的研究

利用两种不同的变换,获得了一类非线性耦合标量场方程的若干类型的精确解析解,其中包括孤子解、奇性孤波解和三角函数解,从而丰富了方程解的内容。这些结论可以应用于其它的非线性方程。此外还纠正了一些文献的部分结论。     

Numerical Continuation Method for Nonlinear System of Scalar and Functional Equations

Computational Mathematics and Mathematical Physics - We propose a numerical continuation method for a nonlinear system that consists of scalar and functional equations. At each parameter step, we...     

非线性耦合标量场方程新的精确行波解

对于具有丰富物理意义和众多应用价值的非线性耦合标量场方程,通过将所求方程约化为初等积分形式,再利用多项完全判别系统对被积函数中的多项式的根进行分类,得到该方程的丰富的精确解,其中包含有理函数型解,孤波解,三角函数型周期解,椭圆函数型周期解,这其中有许多是新解.这一方法是非常简洁而又有力的,对于椭圆方程,它能得到所有可能的解,对于更高阶方程,与其它方法相比,多项式完全判别系统方法是更有力的.     

Optimal Transport for the System of Isentropic Euler Equations

We introduce a new variational time discretization for the system of isentropic Euler equations. In each timestep the internal energy is reduced as much as possible, subject to a constraint imposed by a new cost functional that measures the deviation of particles from their characteristic paths.     

Limit Sets for Multidimensional Nonlinear Transport Equations

We propose a hyperbolic generalisation of the well-known reaction diffusion equation and study the structure of its ω-limit sets. Under a dissipativity condition on the nonlinearity, we show quasi-convergence of the flow, saying that the ω-limit set of any orbit is non-empty, compact, connected, and contained in the set of equilibrium solutions.     

Fourier Multipliers and Littlewood‐Paley for modulation spaces

In this paper we have studied Fourier multipliers and Littlewood‐Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space into itself possesses an l₂‐valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical ‐spaces.     

Triple intersection numbers for the Paley graphs

We give a tight bound for the triple intersection numbers of Paley graphs. In particular, we show that any three vertices have a common neighbor in Paley graphs of order larger than 25.     

On the topology of metric f – K -contact manifolds

We observe that the class of metric f–K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f–K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.