Van Vleck Assistant Professor
803 Van Vleck, 480 Lincoln Dr
Department of Mathematics
University of Wisconsin, Madison, WI 53706
email: mtran23 at wisc dot edu
I have been carrying on interdisciplinary researches, focusing on the following topics:
Main THEME A: Kinetic and Dispersive Equations – Mathematical Physics
A. Wave (weak) turbulence and quantum kinetics:
In continuum mechanics, wave turbulence (WT) is a set of nonlinear waves deviated far from thermal equilibrium. Quantum kinetic (QK) theory is developed to study the dynamics of finite temperature Bose-Einstein condensate systems. Both theories use similar Boltzmann equations, whose solutions can describe the behavior of -norm of the solutions of nonlinear dispersive equations (cubic NLS, water waves, etc.) (see Tao Blog or this AIM workshop for an overview). However, very little is known rigorously in this field. In my works, I have:
- provided the first existence and uniqueness results for the Boltzmann equation derived from capillary water waves;
- provided the first existence and uniqueness results for the Boltzmann equation derived from the primitive equation in oceanography;
- improved the local theory by Escobedo-Velazquez (Invent Math’15) by adding the condensate-thermal cloud collision operator to the equation, and proving global existence instead of blow up results (in this work, the more general Bogoliubov dispersion relation is used instead of just the classical one);
- provided existence, uniqueness, creation and propagation of moments for solutions of the quantum Boltzmann equation, in which the cut-off assumption on the kernel Arkeryd-Nouri (CMP’12) is removed;
- made the connection between the Global Attractor Conjecture and the convergence to equilibrium problem of WT-QK;
- contributed to the derivation of the QK equations using quantum field theory;
- studied the coupling problem of QK and nonlinear Schrodinger equations, using normal form transformation;
- proved that the solutions of the QK equation are bounded from below by Gaussians;
- proved the convergence to equilibrium with algebraic decay rates;
- corrected the hydrodynamics limits of Allemand’s thesis (Paris 6, 2010);
- improved the local theory by Escobedo-Velazquez (Memoirs AMS’15) from 1D to 3D and from classical to general dispersion relations.
B. Classical Kinetic Theory. Kinetic theory describes a gas as submicroscopic particles in constant rapid motion. The Boltzmann equation describes the statistical behavior of a large number of particles away from equilibrium. The Kolmogorov-Fokker-Planck equation describes the time evolution of the probability density function of the velocity of a single particle under the influence of drag forces. The Goldsten-Taylor model describes the behavior of a gas composed of two kinds of particles moving parallel to the $x$-axis with different constant speeds.
- Boltzmann equation: I have used nonlinear approximation theory for the numerical resolution of the homogeneous equation;
- Kolmogorov-Fokker-Planck equation: I have designed a structure preserving scheme, that preserves the long time dynamics of the solution of the homogeneous equation numerically;
- Goldsten-Taylor model: I have resolved a conjecture by Desvillettes-Salvarani about the exponential convergence to equilibrium of solutions to the system.
MAIN THEME B: Scientific Computing-Computational Sciences and Uncertainty QuantiFIcation-Sensitivity Analysis
A.Uncertainty quantification (UQ) is the science of quantitative characterization and reduction of uncertainties in computational and real world applications. Sensitivity analysis (SA) is the study of how the uncertainty in the output of a model can be apportioned to sources of uncertainty in its inputs.
- I have done the first rigorous sensitivity analysis for conservation laws.
B. Domain decomposition methods (DDMs) are parallel computing techniques used to solve a boundary value problem by splitting it into smaller boundary value problems on subdomains. Hybridized discontinuous Galerkin (HDG) methods are techniques combining features of the finite element and the finite volume frameworks.
- I have presented a scalable iterative solver for high-order HDG discretizations of linear partial differential equations. It is well suited for computing systems with massive concurrences.
- I have developed a new machinery to study the convergence problem of both classical and optimized DDMs. I have also studied the applications of DDMs to stochastic differential equations and financial maths; kinetic equations; control theory.