A Super Camassa–Holm Equation with N‐Peakon Solutions

A super Camassa–Holm equation with peakon solutions is proposed, which is associated with a 3 × 3 matrix spectral problem with two potentials. With the aid of the zero‐curvature equation, we derive a hierarchy of super Harry Dym type equations and establish their Hamiltonian structures. It is shown that the super Camassa–Holm equation is exactly a negative flow in the hierarchy and admits exact solutions with N peakons. As an example, exact 1‐peakon solutions of the super Camassa–Holm equation are given. Infinitely many conserved quantities of the super Camassa–Holm equation and the super Harry Dym type equation are, respectively, obtained.     

On Solutions to a Two‐Component Generalized Camassa‐Holm Equation

We consider a two‐component Camassa–Holm system which arises in shallow water theory. We analyze a wave breaking mechanism and the global existence of solutions. First, we discuss the local well posedness and a blow up mechanism, then establish some new blow up criteria for this system formulated either on the line or with space‐periodic initial conditions. Finally, the existence of global solutions is analyzed.     

A New Integrable Equation with Peakon Solutions

We consider a new partial differential equation recently obtained by Degasperis and Procesi using the method of asymptotic integrability; this equation has a form similar to the Camassa–Holm shallow water wave equation. We prove the exact integrability of the new equation by constructing its Lax pair and explain its relation to a negative flow in the Kaup–Kupershmidt hierarchy via a reciprocal transformation. The infinite sequence of conserved quantities is derived together with a proposed bi-Hamiltonian structure. The equation admits exact solutions as a superposition of multipeakons, and we describe the integrable finite-dimensional peakon dynamics and compare it with the analogous results for Camassa–Holm peakons.     

Global Solutions of Two‐Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L² ∩ L^∞ and the initial velocity is small in L² and bounded in L^p for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd‐B model, the additional assumption on the initial velocity being bounded in L^p for some p > 2 may due to techniques we employed. The smallness assumption on the L² norm of the initial velocity is, however, natural for global well‐posedness. One of the key observations in the paper is that the velocity and the “ effective viscous flux” are sufficiently regular for positive time. The regularity of leads to a new approach for the pointwise estimate for the deformation gradient without using L^∞ bounds on the velocity gradients in spatial variables. © 2015 Wiley Periodicals, Inc.     

Convergence of the Two‐Dimensional Dynamic Ising‐Kac Model to

The Ising‐Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighborhood of radius γ ^{? 1} for around its base point. We study the Glauber dynamics for this model on a discrete two‐dimensional torus for a system size and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse‐grained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation. This equation is the dynamic version of the quantum field theory, which is formally given by a reaction‐diffusion equation driven by an additive space‐time white noise. It is well‐known that in two spatial dimensions such equations are distribution valued and a Wick renormalization has to be performed in order to define the nonlinear term. Formally, this renormalization corresponds to adding an infinite mass term to the equation. We show that this need for renormalization for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.© 2016 by the authors. Communications on Pure and Applied Mathematics is published by Wiley Periodicals, Inc., on behalf of the Courant Institute of Mathematics.     

Two‐dimensional chemotaxis models with fractional diffusion

In this paper we study the Cauchy problem for the fractional diffusion equation u_t + (?Δ)^α/2u=?·(u?(Δ^?1u)), generalizing the Keller–Segel model of chemotaxis, for the initial data u₀ in critical Besov spaces ?(?²) with r∈[1, ∞], where 1<α<2. Making use of some estimates of the linear dissipative equation in the frame of mixed time–space spaces, the Chemin ‘mono‐norm method,’ Fourier localization technique and the Littlewood–Paley theory, we obtain a local well‐posedness result. We also consider analogous ‘doubly parabolic’ models. Copyright © 2008 John Wiley & Sons, Ltd.     

Two‐coloring random hypergraphs

A 2‐coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. Let H=H(k, n, p) be a random k‐uniform hypergraph on a vertex set V of cardinality n, where each k‐subset of V is an edge of H with probability p, independently of all other k‐subsets. Let $ m = p{{n}\choose{k}}$ denote the expected number of edges in H. Let us say that a sequence of events ?_n holds with high probability (w.h.p.) if lim_n→∞Pr[?_n]=1. It is easy to show that if m=c2^kn then w.h.p H is not 2‐colorable for c>ln 2/2. We prove that there exists a constant c>0 such that if m=(c2^k/k)n, then w.h.p H is 2‐colorable. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 249–259, 2002     

Two‐cardinal diamond star

Our main results are: (A) It is consistent relative to a large cardinal that holds but fails. (B) If holds and are two infinite cardinals such that and λ carries a good scale, then holds. (C) If are two cardinals such that κ is λ‐Shelah and , then there is no good scale for λ.     

A Stochastic Analysis of Some Two‐Person Sports

We consider two‐person sports where each rally is initiated by a server, the other player (the receiver) becoming the server when he/she wins a rally. Historically, these sports used a scoring based on the side‐out scoring system, in which points are only scored by the server. Recently, however, some federations have switched to the rally‐point scoring system in which a point is scored on every rally. As various authors before us, we study how much this change affects the game. Our approach is based on a rally‐level analysis of the process through which, besides the well‐known probability distribution of the scores, we also obtain the distribution of the number of rallies. This yields a comprehensive knowledge of the process at hand, and allows for an in‐depth comparison of both scoring systems. In particular, our results help to explain why the transition from one scoring system to the other has more important implications than those predicted from game‐winning probabilities alone. Some of our findings are quite surprising, and unattainable through Monte Carlo experiments. Our results are of high practical relevance to international federations and local tournament organizers alike, and also open the way to efficient estimation of the rally‐winning probabilities.     

Two‐dimensional fractal Burgers equation with step‐like initial conditions

We consider the two‐dimensional convection–diffusion equation with a fractional Laplacian, supplemented with step‐like initial conditions. We show that the large time behavior of solutions to this IVP is described either by rarefaction waves, or diffusion waves, or suitable self‐similar solutions, depending on the order of the fractional dissipation and on a direction of a convective nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.     

Remarks on a Strongly Nonlinear Model for Two‐Layer Flows with a Top Free Surface

We revisit in this paper the strongly nonlinear long wave model for large amplitude internal waves in two‐layer flows with a free surface proposed by Choi and Camassa [1] and Barros et al. [2]. Its solitary‐wave solutions were the object of the work by Barros and Gavrilyuk [3], who proved that such solutions are governed by a Hamiltonian system with two degrees of freedom. A detailed analysis of the critical points of the system is presented here, leading to some new results. It is shown that conjugate states for the long wave model are the same as those predicted by the fully nonlinear Euler equations. Some emphasis will be given to the baroclinic mode, where interfacial waves are known to change polarity according to different values of density and depth ratios. A critical depth ratio separates these two regimes and its analytical expression is derived directly from the model. In addition, we prove that such waves cannot exist throughout the whole range of speeds.     

Two‐dimensional adaptive Fourier decomposition

One‐dimensional adaptive Fourier decomposition, abbreviated as 1‐D AFD, or AFD, is an adaptive representation of a physically realizable signal into a linear combination of parameterized Szegö and higher‐order Szegö kernels of the context. In the present paper, we study multi‐dimensional AFDs based on multivariate complex Hardy spaces theory. We proceed with two approaches of which one uses Product‐TM Systems; and the other uses Product‐Szegö Dictionaries. With the Product‐TM Systems approach, we prove that at each selection of a pair of parameters, the maximal energy may be attained, and, accordingly, we prove the convergence. With the Product‐Szegö dictionary approach, we show that pure greedy algorithm is applicable. We next introduce a new type of greedy algorithm, called Pre‐orthogonal Greedy Algorithm (P‐OGA). We prove its convergence and convergence rate estimation, allowing a weak‐type version of P‐OGA as well. The convergence rate estimation of the proposed P‐OGA evidences its advantage over orthogonal greedy algorithm (OGA). In the last part, we analyze P‐OGA in depth and introduce the concept P‐OGA‐Induced Complete Dictionary, abbreviated as Complete Dictionary. We show that with the Complete Dictionary P‐OGA is applicable to the Hardy H² space on 2‐torus. Copyright © 2016 John Wiley & Sons, Ltd.     

On Multi‐component Ermakov Systems in a Two‐Layer Fluid: Integrable Hamiltonian Structures and Exact Vortex Solutions

By introducing an elliptic vortex ansatz, the 2+1‐dimensional two‐layer fluid system is reduced to a finite‐dimensional nonlinear dynamical system. Time‐modulated variables are then introduced and multicomponent Ermakov systems are isolated. The latter is shown to be also Hamiltonian, thereby admitting general solutions in terms of an elliptic integral representation. In particular, a subclass of vortex solutions is obtained and their behaviors are simulated. Such solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is proved that the Hamiltonian system is equivalent to the stationary nonlinear cubic Schrödinger equations coupled with a Steen‐Ermakov‐Pinney equation.     

Isoperimetry in Two‐Dimensional Percolation

We study isoperimetric sets, i.e., sets with minimal boundary for a prescribed volume, on the unique infinite connected component of supercritical bond percolation on the square lattice. In the limit of the volume tending to infinity, properly scaled isoperimetric sets are shown to converge (in the Hausdorff metric) to the solution of an isoperimetric problem in ?² with respect to a particular norm. As part of the proof we also show that the anchored isoperimetric profile as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the square lattice. © 2015 Wiley Periodicals, Inc.     

A New Class of Two‐Layer Green–Naghdi Systems with Improved Frequency Dispersion

We introduce a new class of Green–Naghdi type models for the propagation of internal waves between two (1 + 1)‐dimensional layers of homogeneous, immiscible, ideal, incompressible, and irrotational fluids, vertically delimited by a flat bottom and a rigid lid. These models are tailored to improve the frequency dispersion of the original bi‐layer Green–Naghdi model, and in particular to manage high‐frequency Kelvin–Helmholtz instabilities, while maintaining its precision in the sense of consistency. Our models preserve the Hamiltonian structure, symmetry groups, and conserved quantities of the original model. We provide a rigorous justification of a class of our models thanks to consistency, well‐posedness, and stability results. These results apply in particular to the original Green–Naghdi model as well as to the Saint–Venant (hydrostatic shallow water) system with surface tension.     

Two‐dimensional stationary Navier–Stokes equations with 4‐cyclic symmetry

This paper is concerned with the stationary Navier–Stokes equation in the whole plane and in the two–dimensional exterior domain invariant under the action of the cyclic group of order 4, and gives a condition on the potentials yielding the external force, and on the boundary value, sufficient for the unique existence of a small solution equivariant with respect to the aforementioned cyclic group.     

Two‐level discretization of the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity

The r‐Laplacian has played an important role in the development of computationally efficient models for applications, such as numerical simulation of turbulent flows. In this article, we examine two‐level finite element approximation schemes applied to the Navier‐Stokes equations with r‐Laplacian subgridscale viscosity, where r is the order of the power‐law artificial viscosity term. In the two‐level algorithm, the solution to the fully nonlinear coarse mesh problem is utilized in a single‐step linear fine mesh problem. When modeling parameters are chosen appropriately, the error in the two‐level algorithm is comparable to the error in solving the fully nonlinear problem on the fine mesh. We provide rigorous numerical analysis of the two‐level approximation scheme and derive scalings which vary based on the coefficient r, coarse mesh size H, fine mesh size h, and filter radius δ. We also investigate the two‐level algorithm in several computational settings, including the 3D numerical simulation of flow past a backward‐facing step at Reynolds number Re = 5100. In all numerical tests, the two‐level algorithm was proven to achieve the same order of accuracy as the standard one‐level algorithm, at a fraction of the computational cost. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Peakon,pseudo‐peakon,cuspon and smooth solitons for a nonlocal Kerr‐like media

This paper presents all possible exact explicit peakon, pseudo‐peakon, cuspon and smooth solitary wave solutions for a nonlocal Kerr‐like media. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. We present peakon, pseudo‐peakon, cuspon soliton solution in an explicit form. We also have obtained smooth soliton. Mathematical analysis and numeric graphs are provided for those soliton solutions of the nonlocal Kerr‐like media. Copyright © 2016 John Wiley & Sons, Ltd.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017