Mathematics Home / 2013 Clifford Lectures

**Clifford Lecturer: Peter Constantin (Princeton University)**

**Main Lectures:**

- Problems with fluids: an introduction
- Regularity, long time behavior and absence of anomalous dissipation in 2D SQG
- Stochastic, damped, driven Euler equations
- Complex Fluids

**Invited Speakers:**

- Gautam Iyer (Carnegie Mellon University)
- Alex Kiselev (University of Wisconsin at Madison)
- Nader Masmoudi (CIMS, New York University Anna Mazzucato (Penn State University)
- Dong Li (University of British Columbia)
- Weiran Sun (Simon Fraser University)
- Edriss Titi (University of California at Irvine, Weizmann Institute of Science)
- Jiahong Wu (Oklahoma State University)
- Andrej Zlatos (University of Wisconsin at Madison)

**Outside Participants:**

• Mihaela Ignatova, Stanford University

• Aseel Farhat, Indiana University

• Yao Yao, University of Wisconsin at Madison

• Adam Larios, Texas A&M University

• Kyudong Choi, University of Wisconsin at Madison

• Jacob Bedrossian, New York University

• Andrei Tarfulea, Princeton University

• Lizheng Tao, Oklahoma State University

• Qingtian Zhang, Penn State University

• Yajie Zhang, Penn State University

• Chris Henderson, Stanford University

• Xiaoqian Xu, University of Wisconsin at Madison

• Tau Lim, University of Wisconsin at Madison

• Tarek Elgindi, New York University

• Xiaoyi Zhang, University of Iowa

• Shawn Walker, Louisiana State University

• Barbara Keyfitz, Ohio State University

• Yixiang Wu, University of Louisiana at Lafayette

• Tam Do, University of Wisconsin at Madison

• Anthony Polizzi, Louisiana State University

• Nguyen Phuc, Louisiana State University

• Mimi Dai, University of Illinois at Chicago

• Jihong Hu, University of Pittsburgh

• Rongfang Zhang, University of Pittsburgh

• Huaqiao Wang, University of Pittsburgh

• Tural Sadigov, Indiana University

**Local Participants:**

- Kasha Saxton, Loyola University
- Ralph Saxon, University of New Orleans
- Ricardo Cortez, Tulane University
- Lisa Fauci, Tulane University
- Mac Hyman, Tulane University
- Alex Kurganov, Tulane University
- Xuefeng Wang, Tulane University
- Kun Zhao, Tulane University
- Christina Hamlet, Tulane University
- Julie Simons, Tulane University
- Jacek Wrobel, Tulane University
- Carrie Manore, Tulane University
- Kyle Hickman, Tulane University
- Huicong Li, Tulane University
- Tian Xiang, Tulane University
- Qiang Yang, Tulane University
- Franz Hoffmann, Tulane University
- Elham Ahmadi, Tulane University
- Shanshan Jiang, Tulane University
- Jianjun Huang, Tulane University
- Yuanzhen Cheng, Tulane University
- Shumo Cui, Tulane University
- Zhuolin Qu, Tulane University
- Tong Wu, Tulane University

**Lecture 1. Problems with fluids: an introduction.**

**Abstract**

I will describe some of the basic problems driving the mathematical analysis of incompressible fluid equations. They will include of course well-known regularity issues, the Onsager conjecture, and questions regarding models of complex fluids.

**Lecture 2. Regularity, long time behavior and absence of anomalous dissipation in 2D SQG.**

**Abstract**

This lecture will be devoted to the critical dissipative SQG equation, and will present recent results concerning the regularity of solutions, the absence of remanent dissipation for the viscous, forced equation, and the finite dimensionality of the long time behavior.

**Lecture 3. Damped, stochastic, driven Euler equations.**

**Abstract**

I will present a proof of unique ergodicity for the system consisting of a stochastically forced, fractionally dissipated 2D incompressible Euler equations.

**Lecture 4. Complex Fluids.**

**Abstract**

This lecture will address results concerning kinetic models of complex fluids and their closures. I will present proofs of global regularity and asymptotic stability for some models, and describe some of the many remaining challenges.

**Mixing of passive scalars by incompressible enstrophy-constrained flows**

**Abstract**

Consider a diffusion-free passive scalar $\theta$ being mixed by an incompressible flow $u$ on the torus $\mathbb T^d$. We study how well this scalar can be mixed under an enstrophy constraint on the advecting velocity field. Our main result shows that the mix-norm ($\|\theta(t)\|_{H^{-d/2}}$) is bounded below by an exponential function of time. The exponential decay rate is morally the measure of the support of the initial data, and agrees with both physical intuition and numerical simulations. The main idea behind our proof is to use the notion of ``mixed to scale $\delta$'' and recent work of Crippa and DeLellis towards the proof of Bressan's rearrangement cost conjecture.

**Small scale creation in 2D Euler equation for ideal flow**

**Abstract**

The global existence of smooth solutions for 2D Euler equation in smooth bounded domain has been known since 1933. The equation is in some sense critical, as the needed estimates barely close and the upper bound on the possible growth of the gradient of vorticity is a double exponential in time. There has been much research on whether this bound is sharp, but until recently the gap between the best infinite time growth example and double exponential remained huge. I will provide a review of the subject; then I will discuss a recent construction which gives a solution for 2D Euler in a disk where the growth in vorticity gradient is indeed a double exponential in time. This proves sharpness of the upper bound and illustrates an important role of boundaries in creation of small scales.

**Norm inflation for incompressible Euler in critical spaces**

**Abstract**

I will discuss some recent results on the norm inflation phenomena for incompressible Euler equations in critical spaces.

**Nonlinear inviscid damping in 2D Euler**

**Abstract**

We prove the global asymptotic stability of shear flows close to planar Couette flow in the 2D incompressible Euler equations. Specifically, given an initial perturbation of the Couette flow which is small in a suitable regularity class we show that the velocity converges strongly in L2 to another shear flow which is not far from Couette. This strong convergence is usually referred to as "inviscid damping" and is roughly analogous to Landau damping in the Vlasov Poisson equations which was recently proved by Mouhot and Villani. We will also present an alternative proof of the Landau damping. This is a joint work with Jacob Bedrossian.

**Boundary layers in non-linear pipe and channel flows**

**Abstract**

I will present rigorous results about viscous boundary layers for certain non-linear incompressible flows in pipes and channels. A viscous boundary layer is formed near walls due to friction in flows at high Reynolds numbers, where gradients of velocity are possibly large and vorticity is created. I will justify Prandtl approximation for these flows and study concentration of vorticity at the boundary in the limit of vanishing viscosity under classical no-slip boundary conditions.

**Moment closures for kinetic equations**

**Abstract**

Kinetic equations are widely used to describe the evolution of particle density functions. Computationally these equations can be expensive when there are many particles involved. Moment closures are developed as reductions of kinetic equations. These equations can be viewed as macroscopic models which govern various moments of density functions. A physically and computationally meaningful question is how to de-rive compatible boundary conditions for moment closures when given certain boundary conditions for the underlying kinetic equation. In this talk we give a formal derivation of boundary conditions for moment systems that are derived from linear or linearized kinetic equations with incoming data. This derivation can be applied to various types of kinetic equations such as the linear neutron transport equations and the linearized Boltzmann equation. For simple stationary systems we will show the well-posedness and accuracy of the moment systems with the so-derived boundary conditions.

**Finite Number of Determining Parameters for the Navier-Stokes Equations with Applications into Feedback Control and Data Assimilations**

**Abstract**

In this talk we will implement the notion of finite number of determining parameters for the long-time dynamics of the Navier-Stokes equations (NSE), such as determining modes, nodes, volume elements, and other determining interpolants, to design finite-dimensional feedback control for stabilizing their solutions. The same approach is found to be applicable for data assimilations. In addition, we will show that the long-time dynamics of the NSE can be imbedded in an infinite-dimensional dynamical system that is induced by an ordinary differential equations, named determining form, which is governed by a globally Lipschitz vector field. The NSE are used as an illustrative example, and all the above mentioned results hold also to other dissipative evolution equations.

**Fluid Partial Differential Equations with Partial or Fractional Dissipation**

**Abstract**

This talk presents recent results on the global regularity problem concerning several PDEs modeling fluids when only partial dissipation or fractional dissipation is involved. Attention is focused on the 2D Boussinesq equations and the 2D magneto-hydrodynamic (MHD) equations. The Boussinesq equations concerned here model geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower dimensional model of the 3D hydrodynamics equations. We review recent results on various cases of partial dissipation and presents most recent developments on the 2D Boussinesq equations with fractional dissipation. The MHD equations model electrically conducting fluids such as plasmas. The MHD equations can be difficult to analyze due to the nonlinear coupling between the induction equation and the Naver-Stokes equations with the Lorentz force. Global regularity results for several partial and fractional cases will be summarized. Especially small global solutions with velocity dissipation or damping will be described.

**Reactive Processes in Inhomogeneous Media**

**Abstract**

We study fine details of spreading of reactive processes (e.g., combustion) in multi-dimensional inhomogeneous media. One typically observes a transition from one equilibrium (e.g., unburned fuel) to another (e.g., burned fuel) to happen on short spatial as well as temporal scales. We demonstrate that this phenomenon also occurs in one of the simplest models for reactive processes, reaction-diffusion equations with ignition reaction functions (as well as with some monostable and bistable reaction functions, in a slightly weaker form), under very general hypotheses. Specifically, we show that in up to three spatial dimensions, the width (both in space and time) of the zone where reaction occurs stays uniformly bounded in time for some fairly general classes of initial data, and this bound even becomes independent of the initial datum as well as the reaction function, after an initial time interval. Such results have recently been obtained in one spatial dimension but were unknown in higher dimensions. As one indication of the added difficulties, we also show that three dimensions is indeed the borderline case, and the result is false for general inhomogeneous media in four and more dimensions.