### Dr Bin Cheng

### Biography

Dr. Cheng was born and raised in China. He earned a Bachelor's degree in Mathematics from the Peking University (PKU) at Beijing, China in 2001, went on to study in the University of California at Los Angeles (UCLA), USA where he earned a Master's degree in Applied Mathematics in 2003, and in the University of Maryland, College Park (UMD), USA where he earned his Ph.D. in Applied Mathematics and Scientific Computing in 2007.

Dr. Cheng then took on a Hildebrandt Assistant Professorship (postdoctoral) at the University of Michigan, Ann Arbor, USA from 2007 to 2010, and a Research Assistant Professorship (postdoctoral) at the Arizona State University, Tempe, USA from 2010 to 2012.

Dr. Cheng joined the University of Surrey at Guildford, UK in January, 2013. He currently holds a Senior Lectureship (roughly the UK equivalent of tenured assistant professor) in the Department of Mathematics. His research interests sit in the general areas of applied mathematics and mathematical physics, especially in the analysis of nonlinear PDEs and applications to large-scale atmospheric and oceanic circulations. But he is curious just about everything.

Dr. Cheng has taught university-level mathematics for over 10 years, and has also delivered numerous seminars and colloquia. He has much experience in communicating mathematics to others and always enjoys doing so.

His personal website is at http://personal.maths.surrey.ac.uk/st/bc0012/

### Research

### Research interests

I am the PI of the project "Mathematical analysis of ‘near resonance’ in the physical world of finiteness" funded by a Leverhulme Research Project Grant (RPG-2017-098), 2017-2021.

I am also a co-I of the project "On the way to the asymptotic limit: mathematics of slow-fast coupling in PDEs" funded by an EPSRC Standard Research Grant (EP/R029628/1), 2018-2021.

### Supervision

# Postgraduate research supervision

### My publications

### Publications

Despite many efforts in this direction, the IMs in the case when the spectral gap conditions are violated remain a mystery especially in the case of parabolic PDEs. On the one hand, there is a number of interesting classes of such equations where the existence of IMs is established without the validity of the spectral gap conditions and, on the other hand there were no examples of dissipative parabolic PDEs where the non-existence of an IM is rigorously proved.

The main aim of this thesis is to bring some light on this mystery by the comprehensive study of three model examples of parabolic PDEs where the spectral gap conditions are not satisfied, namely, 1D reaction-diffusion-advection (RDA) systems (see Chapter 3), the 3D Cahn-Hilliard equation on a torus (see Chapter 4) and the modified 3D Navier-Stokes equations (see Chapter 5). For all these examples the existence or non-existence of IM was an open problem.

As shown in Chapter 3, the existence or non-existence of an IM for RDA systems strongly depends on the boundary conditions. In the case of Dirichlet or Neumann boundary conditions, we have proved the existence of an IM using a specially designed non-local in space diffeomorphism which transforms the equations to the new ones for which the spectral gap conditions are satisfied. In contrast to this, in the case of periodic boundary conditions, we construct a natural example of a RDA system which does not possess an IM.

In Chapters 4 and 5 we develop an extension of the so-called spatial averaging principle (SAP) (which has been suggested by Sell and Mallet-Paret in order to treat scalar reaction-diffusion equation on a 3D torus) to the case of 4th order equations where the nonlinearity loses smoothness (the Cahn-Hilliard equation) as well as

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 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.

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.

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.