Description of Dr. Xiaoming Wang's current research

   My research interest lies in modern applied mathematics, especially problems related to fluid dynamics, groundwater research, geophysical fluid dynamics and turbulence. The mathematical tools that I use in my research include Partial Differential Equations, variational and perturbative methods, dynamical systems, stochastic analysis, numerical analysis and scientific computing. A distinctive feature of my work is the combination of rigorous mathematics with genuine physical applications.
   Please refer to my publications page for some of the papers mentioned below.
   Prospective students are encouraged to contact me directly.

Long time statistical properties of large chaotic/turbulent dissipative systems
Many dissipative dynamical systems arising in physical applications possess very complex behaviors with abundant instability and sensitive dependence on initial data and parameters. It is well-known that the statistical properties of these kinds of systems are much more important, physically relevant and stable than single trajectories. For instance, Kolmogorov's turbulence theory is formulated in terms of the statistical properties of the Navier-Stokes system. Many times, the physical system under investigation reaches a statistical equilibrium over a long time. The long-time statistical behavior of the system is then encoded in the associated invariant measures or stationary statistical solutions of the system. Part of my research focuses on the analysis and approximation of these long time statistical properties for large chaotic/turbulent systems.
Several of my papers are devoted to the issue of the stability of these long-time statistical properties on certain parameters within the model. This can be viewed as model validation issue at the long-time statistical property level, since many simplified models are derived by setting certain parameter to zero or infinity. Several general results on the continuity of long-time statistical behavior (invariant measures) under regular and singular perturbation of the underlying models have been derived. My collaborator and I also worked on the physically important rate of heat transport in the vertical direction - a specific long-time statistical property - under the Rayleigh-Benard setting. Several physically meaningful and mathematical interesting results have been derived in a few recent works. In particular, we are able to show that the infinite Prandtl number model for convection is a good approximation of the Rayleigh-Benard convection at large Prandtl number in the long-time statistical sense, although the perturbation itself is a singular one. We have also derived rigorous bounds on the rate of heat transport in the vertical direction for convection at large Prandtl number that are consistent with the best known estimates on the rate of heat transport for the infinite Prandtl number model. Long-time averaged energy dissipation rate per unit mass, another specific long-time statistical property, plays a central role in conventional theory for turbulence. I was able to derive physically interesting results that are consistent with Kolmogorov's heuristically physical arguments on long-time averaged energy dissipation rate per unit mass.
Due to the presumed complexity of the dynamics for many dissipative systems, most physically interesting stationary statistical properties need to be calculated using numerical methods in the generic case. Even under the ergodicity assumption, it is not at all clear that classical numerical schemes, which provide accurate approximation on finite time interval, will remain meaningful for stationary statistical properties (long time properties) since small errors will be amplified and accumulated over a long time except in the case that the underlying dynamics are asymptotically stable (where statistical approach is not necessary as there is no chaos). Therefore, it is a challenge of great importance to develop numerical methods that are able to capture stationary statistical properties of infinite dimensional complex dynamical systems. Several of my recent papers are devoted to this topic. In particular, I have established a Lax type criterion that guarantees the convergence of long-time statistical properties under first order time discretization. This Lax type criterion provides guidelines on how to design numerical algorithms that are able to capture long-time statistical properties. Together with collaborators, I have also developed several first and second order accurate in time, fast, and efficient numerical schemes for the Navier-Stokes system and its variations. I am currently developing fully discretized efficient schemes that are able to capture the long time statistical properties of the underlying dissipative systems.

Fluid flows and in karstic geometry
  • Flow in fluid saturated karst aquifers
    • Karst is a common type of landscape formed by the dissolution of layers of soluble bedrock, usually including carbonate rock, limestone, and dolomite. Karst regions often contain karst aquifers, which are important sources of potable water. For example, about 90% of the fresh water used in Florida comes from karst aquifers. Clearly, the study of karst aquifers is of great importance, especially because they are seriously threatened by contamination. A karst aquifer, in addition to a porous limestone or dolomite matrix, typically has large cavernous conduits that are known to have great impact on groundwater flow and contaminant transport within the aquifer. During high-rain seasons, the water pressure in the conduits is larger than that in the ambient matrix so that conduit borne contaminants can be driven into the matrix. During dry seasons, the pressure differential reverses and contaminants long sequestered in the matrix can be released into the free flow in the conduits and exit through, e.g., springs and wells, into surface water systems. Therefore, an understanding of the interaction between the free flow in the conduits and the Darcy flow in the matrix is crucial to the study of groundwater flows and contaminant transport in karst regions. The mathematical study of flows in karst aquifers is a well-known challenge due to the coupling of the flow in the conduits and the flow in the surrounding matrix, the complex geometry of the network of conduits, the vastly disparate spatial and temporal scales, the strong heterogeneity of the physical parameters, and the huge associated uncertainties in the data.
    I, together with my collaborators, have made several contributions to flows in fluid saturated karst aquifers. First, we have established the mathematical validity of the Stokes-Darcy model with the classical Beavers-Joseph matrix-conduit interface boundary condition. Second, I have identified the mathematical ill-posedness of the coupled continuum pipe flow (CCPF) model, which is incorporated in USGS' popular MODFLOW software. Third, we have developed, analyzed and implemented several first and second order in time accurate, unconditionally stable, long-time accurate efficient schemes for the Stokes-Darcy model and simplified coupled continuum pipe flow (CCPF) model. Long-time accurate schemes for systems like the Stokes-Darcy model are highly desirable since the physically interesting phenomena usually occur over a very long time. My collaborators and I are very proud of the algorithms that we developed and analyzed, especially those schemes that are not only long-time accurate in the sense of the existence of uniform in time error bounds, but also highly efficient (only two decoupled simple problems, one Stokes and one Darcy, are involved at each time step). We are now working on the quantification of uncertainty in contaminant transport utilizing the schemes that we developed.
  • Two-phase flow in karstic geometry
    • Multiphase flow phenomena are ubiquitous. Common examples include coupled atmosphere and ocean system (air and water), oil reservoir (water, oil and gas), and cloud and fog (water vapor, water and air). Many multiphase flows are important to us humans and to a lot of engineering processes. Therefore, there is a genuine need to understand various multiphase flow phenomena better.
    In many important applications - such as contaminant transport in an unconfined karst aquifer, oil recovery in karst oil reservoir, proton exchange membrane fuel cell technology, and some cardiovascular modeling - we must deal with multiphase flows in karstic geometry (domain with matrix and conduits or voids). My collaborators and I have made several important contributions to the mathematical investigation of this problem. First, we derived a Cahn-Hilliard-Stokes-Dracy model for two-phase flows in karstic geometry based on Onsager's extremum principle. Second, we established the global in time existence of weak solutions to the Cahn-Hilliard-Stokes-Darcy model. Strong-weak uniqueness result for this model is also derived. Moreover, we have proposed and analyzed a few unconditionally stable and uniquely solvable numerical schemes for this model.
    My collaborators and I also worked on the Cahn-Hilliard-Hele-Shaw model for two phase flows in a Hele-Shaw cell (or porous media). We have established the global in time well-posedness in 2D and a Beale-Kato-Majda type criterion in 3D [8]. Long-time behavior of the model - in particular convergence to steady state - is also thoroughly investigated.
    The theoretical results above form the foundation for further theoretical and numerical investigations that can be used for model validation and prediction.

    Boundary layer and vanishing viscosity
    Boundary layers associated with slightly viscous incompressible fluid flow equipped with the physical no-slip no-penetration boundary condition are of great importance. From the physical point of view, in the absence of body force, it is the vorticity generated in the boundary layer and later advected into the main stream that drives the flow. Indeed, many physical phenomena cannot be explained in a satisfactory fashion without accounting for boundary layer effects (D'Alembert's paradox is one). From the mathematical point of view, the boundary layer problem is a serious challenge, since the slightly viscous fluid equation - the Navier-Stokes system at small viscosity - can be viewed as a singular perturbation of the Euler system that governs the flow of inviscid fluids. Moreover, the leading order singular behavior governed by the so-called Prandtl equation may be ill-posed. Even if the Prandtl boundary-layer system is well posed, the validity of the Prandtl theory ( i.e., the solution of the Navier-Stokes system can be approximated by the solution of the Euler system plus a boundary layer that solves a Prandtl type equation) is not known.
    A number of my publications are related to the study of boundary layers associated with incompressible flows. In particular, I have identified a spectral constraint on the Prandtl solution that ensures the vanity of the Prandtl theory. The spectral constraint can be checked under various physically relevant settings that lead to the verification of the Prandtl theory. For instance, we validated the Prandtl theory for the Navier-Stokes equations when there are non-trivial suction and injection on the boundary. However, the verification of the spectral constraint in the general setting is still an outstanding problem.
    A problem related to the Prandtl boundary layer theory is the vanishing viscosity limit problem of the the Navier-Stokes system. The inviscid Euler's equations emerge after formally setting the viscosity to zero in the Navier-Stokes system. However, the rigorous mathematical justification of the convergence of the solution to the Navier-Stokes system to that of the Euler system under suitable norm (say energy norm) at vanishing viscosity is still elusive. This is commonly referred to as the inviscid limit problem. Validation of the Prandtl boundary layer theory would automatically lead to an affirmative answer to the inviscid limit problem. However, the inviscid limit could be true even if the Prandtl theory is violated, since one can not detect the boundary type structures when using the energy norm. A number of my papers are devoted to this topic. In particular, my collaborator and I have derived several physically interesting necessary and sufficient conditions that guarantee an affirmative answer to the vanishing viscosity limit question. These conditions are consistent with physical intuitions (such as the idea that favorable pressure gradient would prevent separation of boundary layer). My collaborator and I have also discovered that numerical investigation of the inviscid limit problem is a challenge. Indeed, we have shown that if the scheme does not resolve small scales of the order of the viscosity in the direction tangential to the wall, the numerical solutions would converge to the solution of the corresponding Euler system at vanishing viscosity and grid size. This tells us that the intuitive idea of under-resolving the direction parallel to the wall could lead to false confirmation of the inviscid limit.

    Geophysical fluid dynamics
    Fluid motion at large scales is of great importance in geophysical applications. When the scale of the motion is large, one has to take into account the effect of the earth's rotation and/or the stratification of the fluid. An important and interesting common phenomenon associated with large scale fluid motion is the emergence and persistence of large scale coherent structures such as the Great Red Spot on Jupiter. The understanding of this phenomenon is the focus of a series papers and a book by Andy Majda and myself. In these works, we have established the emergence and persistence of large scale coherent structures under the selective decay principle or the influence of small scale random bombardments under appropriate setting. The Cambridge University Press book by Majda and myself summarized both deterministic and empirical statistical mechanics methods for predicting the emergence and persistence of large scale coherent structures with certain basic geophysical fluid dynamics models. The book has been a very useful tool for the community. There are many other problems related to the emergence of large scale coherent structures. For instance Bill Young's simulation of barotropic quasi-geostrophic equation on a beta plane with isotrpic (medium scale) random forcing leads to the emergence of large scale zonal flows in statistical average. The mathematical justification of Bill's numerical result is still missing.
    Model simplification is unavoidable in gaining insight into the understanding of complex behaviors. However, the mathematical rigorous justifications are usually non-trivial. One of my papers with Majda is devoted the the mathematical justification of an important basic geophysical fluid model, the one and one-half layer quasi-geostrophic model from the two-layer models as the ratio of the depths of the two layers (bottom over top) approaches infinity. It turns out that the bottom layer serves as an effective topography for the top layer in the limit of infinite depth ratio. There are many other heuristic model simplifications in geophysical fluid dynamics where mathematically rigorous treatment is still lacking.
    Climate change is of great importance to humankind. Climate can be interpreted as the long-time statistical properties of the underlying climate model. Therefore, one way to study climate change is to investigate the change of long-time statistical properties (such as the temperature) as parameters (such as the level of carbon-dioxide) changes. One of the important tools used in climate change is the fluctuation-dissipation theory, which roughly states that for systems in statistical equilibrium the average mean response to small external perturbations can be calculated through the knowledge of suitable correlation functions of the unperturbed statistical systems. This formulation is particularly appealing in climate study since it only requires the knowledge of today's climate in order to predict the response of the climate to small external perturbations. It is well-known that real climate system involves time-dependent statistical behaviors, such as seasonality. However, the classical setting does not take into account the time-period influence. In a recent work with Majda, we extended the applicability of the classical fluctuation-dissipation theory to a case that allows seasonal changes. We also developed several algorithms on how to approximate the change in long-time statistical behaviors (climate) using the current climate only under time-periodic forcing. This work has led to a few subsequent works by other authors. Additional work in this direction is needed to validate the approach in a mathematically rigorous fashion, and in the development of fast and accurate FDT approximations.

    Numerical methods for models of thin film epitaxy
    Coarsening, i.e., the process by which a group of objects of different sizes transforms into a group consisting of fewer objects with larger average size, is a very common natural phenomenon and has attracted considerable attention recently. The coarsening process usually takes place on a very long time scale for large systems. Therefore it is important to have accurate and efficient time stepping with regards to numerical simulation.
    Many phenomenological macroscopic coarsening processes are energy driven in the sense that the dynamics is governed by the gradient flow of a certain energy functional. One well-known example associated with epitaxial thin film growth is the gradient flow with Ehrlich-Schwoebel type energy. These systems are highly nonlinear and the coarsening process is highly nontrivial, which calls for numerical methods. The design, analysis and implementation of accurate and efficient schemes for thin film epitaxial growth models based on Ehrlich-Schwoebel is the focus of several of my recent papers. In these papers, my collaborators and I developed several unconditionally stable, uniquely solvable, accurate and efficient numerical schemes for the thin film epitaxy models. These novel schemes have been implemented and important physical scaling laws have been verified and/or discovered. There is still a need to develop more efficient and faster numerical algorithms that are able to capture the long time scaling property.

    Long time behavior of dissipative systems
    Many infinite dimensional dynamical systems generated by dissipative evolutionary PDEs have very complex behavior. The fractal dimension of the global attractor, a compact invariant object that attracts all bounded sets in the phase space, is often finite although the system itself lives in an infinite dimensional space (phase space infinite dimensional). The fractal dimension of the global attractor can be interpreted as the degree of freedom of the long-time behavior of the underlying system. In particular, there are heuristic predictions of degrees of freedom for many physically important systems such as the Navier-Stokes equations. For instance, the Kolmogorov's dissipation length in 3D and Kraichnan's dissipation length in 2D all lead to estimates on the degree of freedom for turbulent fluid flows using the heuristic box counting approach. Therefore, a physically interesting question is to derive estimates on the fractal dimension of the global attractor that are consistent with physical/heuristic estimates. Several physical relevant results are available for systems like the 2D Navier-Stokes equations in a periodic box or a channel. However, physically relevant and mathematical rigorous estimates on the degrees of freedom are still missing for many physically important system.