Dr Bin Cheng

We consider Monge-Ampére equations with the right hand side function close to a constant and from a function class that is larger than any Hölder class and smaller than the Dini-continuous class. We establish an upper bound for the modulus of continuity of the solution's second derivatives. This bound depends exponentially on a quantity similar to but larger than the Dini semi-norm. We establish explicit control on the shape of the sequence of shrinking sections, hence revealing the nature of such exponential dependence.

We investigate the Monge-Ampère equation subject to zero boundary value and with a positive right-hand side function assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlinear dependence between the Hölder seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri & Mooney in [FJM16]. Our technique is in part inspired by Jian & Wang in [JW07] which includes using a sequence of so-called sections.

We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid rotating shallow water system with small initial data subject to the zero relative vorticity condition. One of the key steps is a reformulation of the problem into a symmetric quasilinear Klein-Gordon system with quadratic nonlinearity, for which the global existence of classical solutions is then proved with combination of the vector field approach and the normal form method. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity. (C) 2010 Elsevier Inc. All rights reserved.

We study a single column model of moist convection in the atmosphere. We state the conditions for it to represent a stable steady state. We then evolve the column by subjecting it to an upward displacement which can release instability, leading to a time dependent sequence of stable steady states. We propose a definition of measure valued solution to describe the time dependence and prove its existence.

Based on a novel treatment of near resonances, we introduce a new approximation for the rotating stratified Boussinesq system on three-dimensional tori with arbitrary aspect ratios. The rotation and stratification parameters are arbitrary and not equal. We obtain global existence for the proposed nonlinear system for arbitrarily large initial data. This system is sufficiently accurate, with an important feature of coupling effects between slow and fast modes. The key to global existence is a sharp counting of the relevant number of nonlinear interactions. An additional regularity advantage arises from a careful examination of some mixed type interaction coefficients. In a wider context, the significance of our near resonant approach is a delicate balance between the inclusion of more interacting modes and the improvement of regularity properties, compared to the well-studied singular limit approach based on exact resonance.

We improve the recent result of Chae and Tadmor (2008)  [10] proving a one-sided threshold condition which leads to a finite-time breakdown of the Euler–Poisson equations in arbitrary dimension n .

Motivated by recent studies in geophysical and planetary sciences, we investigate the PDE-analytical aspects of time-averages for barotropic, inviscid flows on a fast rotating sphere S-2. Of particular interest is the incompressible Euler equation. We prove that finite-time-averages of solutions stay close to a subspace of longitude-independent zonal flows. The initial data are unprepared and can be arbitrarily far away from this subspace. Our analytical study justifies the global Coriolis effect in the spherical geometry as the underlying mechanism of this phenomenon. We use Riemannian geometric tools including the Hodge theory in the proofs. (C) 2012 Elsevier Masson SAS. All rights reserved.

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We describe a straightforward framework for studying time-averages of dynamical systems whose solutions exhibit fast oscillatory behaviors. Time integration averages out the oscillatory part of the solution that is caused by the large skew-symmetric operator. Then, the time-average of the solution stays close to the kernel of this operator. The key assumption in this framework is that the inverse of the large operator is a bounded mapping between certain Hilbert spaces modular the kernel of the operator itself. This assumption is verified for several examples of time-dependent PDEs.

© 2014 Society for Industrial and Applied Mathematics.This article addresses a fundamental concern regarding the incompressible approximation of fluid motions, one of the most widely used approximations in fluid mechanics. Common belief is that its accuracy is O(ε), where ε denotes the Mach number. In this article, however, we prove an O(ε2) accuracy for the incompressible approximation of the isentropic, compressible Euler equations thanks to several decoupling properties. At the initial time, the velocity field and its first time derivative are of O(1) size, but the boundary conditions can be as stringent as the solid-wall type. The fast acoustic waves are still O(ε) in magnitude, since the O(ε2) error is measured in the sense of Leray projection and more physically, in time-averages. We also show when a passive scalar is transported by the flow, it is O(ε2) accurate pointwise in time to use incompressible approximation for the velocity field in the transport equation.

We prove the global-in-time existence of large-data nite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a transport equation for the probability density function, which models energetic rare ed particles of one species; the incompressible Navier{Stokes system for the bulk uid; and a parabolic evolution equation, involving magnetic di usivity, for the magnetic eld. The physical derivation of our model is given. It is also shown that the weak solution, whose existence is established, has nonincreasing total energy, and that it satis es a number of physically relevant properties, including conservation of the total momentum, conservation of the total mass, and nonnegativity of the probability density function for the energetic particles. The proof is based on a one-level approximation scheme, which is carefully devised to avoid increase of the total energy for the sequence of approximating solutions, in conjunction with a weak compactness argument for the sequence of approximating solutions. The key technical challenges in the analysis of the mathematical model are the nondissipative nature of the Vlasov-type particle equation and passage to the weak limits in the multilinear coupling terms.

We formalise the concept of near resonance for the rotating Navier–Stokes equations, based on which we propose a novel way to approximate the original partial differential equation (PDE). The spatial domain is a three-dimensional flat torus of arbitrary aspect ratios. We prove that the family of proposed PDEs are globally well-posed for any rotation rate and initial datum of any size in any Hs space with $s\geqslant0$. Such approximations retain many more 3-mode interactions, and are thus more accurate, than the conventional exact-resonance approach. Our approach is free from any limiting argument that requires physical parameters to tend to zero or infinity, and is free from any use of small divisors (so that all estimates depend smoothly on the torus's aspect ratios). The key estimate hinges on the counting of integer solutions of Diophantine inequalities rather than Diophantine equations. Using a range of novel ideas, we handle rigorously and optimally challenges arising from the non-trivial irrational functions in these inequalities. The main results and ingredients of the proofs can form part of the mathematical foundation of a non-asymptotic approach to nonlinear, oscillatory dynamics in real-world applications.

With solid-wall boundary condition and ill-prepared initial data, we prove the singular limits and convergence rates of compressible Euler and rotating shallow water equations towards their incompressible counterparts. A major issue is that fast acoustic waves contribute to the slow vortical dynamics at order one and do not damp in any strong sense. Upon averaging in time, however, such a contribution vanishes at the order of the singular parameters (i.e., Mach/Froude/Rossby numbers). In particular, convergence rates of the compressible dynamics, when projected onto the slow manifold, are estimated explicitly in terms of the singular parameters and Sobolev norms of the initial data. The structural condition of a vorticity equation plays a key role in such an estimation as well as in proving singular-parameter-independent life spans of classical solutions.

Time averages are common observables in analysis of experimental data and numerical simulations of physical systems. We will investigate, from the angle of partial differential equation analysis, some oscillatory geophysical fluid dynamics in three dimensions: Navier–Stokes equations in a fast-rotating, spherical shell, and magnetohydrodynamics subject to strong Coriolis and Lorentz forces. Upon averaging their oscillatory solutions in time, interesting patterns such as zonal flows can emerge. More rigorously, we will prove that, when the restoring forces are strong enough, time-averaged solutions stay close to the null spaces of the wave operators, whereas the solutions themselves can be arbitrarily far away from these subspaces.

Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero.

This paper presents a single-column model of moist atmospheric convection. The problem is formulated in terms of conservation laws for mass, moist potential temperature and specific humidity of air parcels. A numerical adjustment algorithm is devised to model the convective adjustment of the column to a statically stable equilibrium state for a number of test cases. The algorithm is shown to converge to a weak solution with saturated and unsaturated parcels interleaved in the column as the vertical spatial grid size decreases. Such weak solutions would not be obtainable via discrete PDE methods, such as finite differences or finite volumes, from the governing Eulerian PDEs. An equivalent variational formulation of the problem is presented and numerical results show equivalence with those of the adjustment algorithm. Results are also presented for numerical simulations of an ascending atmospheric column as a series of steady states. The adjustment algorithm developed in this paper is advantageous over similar algorithms because first it includes the latent heating of parcels due to the condensation of water vapour, and secondly it is computationally efficient making it implementable into current weather and climate models.

We study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [Phys. D, 188 ( 2004), pp. 262-276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t less than or similar to ln(delta(-1)); here delta

Convergence rate estimates are obtained for singular limits of the compressible ideal magnetohydrodynamics equations, in which the Mach and Alfven numbers tend to zero at different rates. The proofs use a detailed analysis of exact and approximate fast, intermediate, and slow modes together with improved estimates for the solutions and their time derivatives, and the time-integration method. When the small parameters are related by a power law the convergence rates are positive powers of the Mach number, with the power varying depending on the component and the norm. Exceptionally, the convergence rate for two components involve the ratio of the two parameters, and that rate is proven to be sharp via corrector terms. Moreover, the convergence rates for the case of a power-law relation between the small parameters tend to the two-scale convergence rate as the power tends to one. These results demonstrate that the issue of convergence rates for three-scale singular limits, which was not addressed in the authors' previous paper, is much more complicated than for the classical two-scale singular limits.

We study classical solutions of one dimensional rotating shallow water system which plays an important role in geophysical fluid dynamics. The main results contain two contrasting aspects. First, when the solution crosses certain thresholds, we prove finite-time singularity formation for the classical solutions by studying the weighted gradients of Riemann invariants and utilizing conservation of physical energy. In fact, the singularity formation will take place for a large class of C 1 initial data whose gradients and physical energy can be arbitrarily small and higher order derivatives should be large. Second, when the initial data have constant potential vorticity, global existence of small classical solutions is established via studying an equivalent form of a quasilinear Klein-Gordon equation satisfying certain null conditions. In this global existence result, the smallness condition is in terms of the higher order Sobolev norms of the initial data.