ENG 
ESP Session S37 - New Developments in Mathematical Fluid Dynamics
Talks
Thursday, July 15, 11:00 ~ 11:25 UTC-3
Analysis of Incompressible Navier--Stokes Equations with Navier Boundary Conditions
Siran Li
New York University Shanghai, China - This email address is being protected from spambots. You need JavaScript enabled to view it.
We are concerned with the mathematical analysis of the Navier--Stokes equations in 3-dimensional domains for incompressible fluid dynamics. The Navier boundary condition, first proposed by Claude-Louis Navier, is a physical correction of the homogeneous zero boundary condition, which states that the tangential components of the stress induced from the fluid particles in the normal directions are proportional to the tangential components of the fluid velocity. We analyse the "geometric" regularity criteria of Navier--Stokes under the Navier boundary condition based on the alignment of vorticity directions, as well as the higher-order estimates for the vanishing viscosity boundary layer expansions.
Joint work with Gui-Qiang G. Chen (University of Oxford, UK) and Zhongmin Qian (University of Oxford, UK).
Thursday, July 15, 11:30 ~ 11:55 UTC-3
Uniqueness and convexity of Whitham’s highest cusped wave I
Alberto Enciso
Instituto de Ciencias Matemáticas, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Whitham’s equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. In this talk we are concerned with non-smooth traveling wave solutions to this equation, which are often referred to as waves of extreme form. Their existence was conjectured by Whitham in 1967 and established by Ehrnström and Wahlén just a few years ago, who proved that there is a monotone traveling wave featuring a cusp of exactly $C^{1/2}$ Hölder regularity at the origin. Our objetive in this talk is to show that there is only one monotone traveling wave of extreme form and that, as widely believed in the community, its profile is in fact convex between crest and trough. This can be understood as the counterpart, in the case of the Whitham equation, of the landmark results on the uniqueness and convexity of traveling water waves of extreme form.
Joint work with Javier Gómez-Serrano (Barcelona and Brown) and Bruno Vergara (Barcelona).
Thursday, July 15, 12:00 ~ 12:25 UTC-3
Uniqueness and convexity of Whitham’s highest cusped wave II
Bruno Vergara
University of Barcelona, Spain - This email address is being protected from spambots. You need JavaScript enabled to view it.
Whitham’s equation is a nonlinear, nonlocal, very weakly dispersive shallow water wave model in one space dimension. In this talk we are concerned with non-smooth traveling wave solutions to this equation, which are often referred to as waves of extreme form. Their existence was conjectured by Whitham in 1967 and established by Ehrnström and Wahlén just a few years ago, who proved that there is a monotone traveling wave featuring a cusp of exactly $C^{1/2}$ Hölder regularity at the origin. Our objetive in this talk is to show that there is only one monotone traveling wave of extreme form and that, as widely believed in the community, its profile is in fact convex between crest and trough. This can be understood as the counterpart, in the case of the Whitham equation, of the landmark results on the uniqueness and convexity of traveling water waves of extreme form.
Joint work with Alberto Enciso (ICMAT, Spain) and Javier Gómez-Serrano (Brown University, USA - University of Barcelona, Spain).
Thursday, July 15, 12:30 ~ 12:55 UTC-3
Singularities for compressible Euler
Tristan Buckmaster
Princeton University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will talk about recent results related to singularity formation for compressible Euler.
Thursday, July 15, 14:00 ~ 14:25 UTC-3
On the stationary and uniformly rotating solutions of the 2D Euler equation
Jaemin Park
Georgia Institute of Technology, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, we discuss whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main result is that for any non-negative smooth stationary vorticity that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation. We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
Joint work with Javier Gomez-Serrano (University of Barcelona, Spain), Jia Shi (Princeton University, United States) and Yao Yao (Georgia Institute of Technology, United States).
Thursday, July 15, 14:30 ~ 14:55 UTC-3
Symmetry in stationary and uniformly-rotating solutions of g-SQG equations
Jia Shi
Princeton University - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk, I will discuss the symmetry in stationary and uniformly rotating solutions of g-SQG. We want to answer whether every stationary/uniformly-rotating solution must be radially symmetric, if the vorticity is compactly supported. If time permits, I will talk about some results for vortex-sheet.
Joint work with Javier Gomez-Serrano(Brown University), Jaemin Park(Georgia Institute of Technology) and Yao Yao(Georgia Institute of Technology).
Thursday, July 15, 15:00 ~ 15:25 UTC-3
On well-posedness of the generalized SQG family in borderline spaces
Vincent Martinez
Hunter College, CUNY, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We review recent results regarding the issue of well-posedness of the family of generalized surface quasi-geostrophic equations, introduced by Chae, Constantin, et. al. 2011, in borderline Sobolev spaces. These equations are a family of active scalar equations, which include the 2D Euler equations as an endpoint and extrapolate beyond with an increasingly singular relation in the constitutive law between the scalar and its advecting velocity. In the most singular regime of velocities, the equations represent a genuinely quasilinear equation, whose coefficients are of higher-order than an arbitrarily small-order dissipative perturbation. We nevertheless show that this obstruction is only apparent due to the underlying commutator structure of the transport nonlinearity. To properly exploit this, however, a new approximation scheme by linear conservation laws is introduced that is able to accommodate this nuanced structure. We also address analogous results in the inviscid setting by considering suitably regularized velocities. In each case, thresholds for the regularizations are identified for which well-posedness can be guaranteed.
Joint work with Michael Jolly (Indiana University) and Anuj Kumar (Indiana University).
Thursday, July 15, 15:30 ~ 15:55 UTC-3
Hypoellipticity and Enhanced Dissipation
Dallas Albritton
Courant Institute of Mathematical Sciences, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the evolution of a passive scalar (for example, a temperature distribution) which is transported by a fluid flow and diffuses: \[ \partial_t f + \boldsymbol{u} \cdot \nabla f = \kappa \Delta f. \] It is well known that shear in the background flow $\boldsymbol{u}$ may interact with the diffusion to cause the solution to decay faster than it would by diffusion alone. This so-called \emph{enhanced dissipation} has been widely studied in the PDE community over the past two decades. We revisit this phenomenon from the new but old perspective of Hormander's classical work on hypoellipticity. This allows us to give short and, in our opinion, transparent proofs of enhanced dissipation in shear flows $\boldsymbol{u} = (b(y),0)$, originally due to Bedrossian and Coti-Zelati. Time permitting, we will also discuss applications to kinetic theory.
Joint work with Rajendra Beekie (Courant Institute of Mathematical Sciences, USA) and Matthew Novack (Courant Institute of Mathematical Sciences, USA).
Friday, July 16, 11:00 ~ 11:25 UTC-3
Bounds on the heat transfer rate via passive advection
Gautam Iyer
Carnegie Mellon University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In heat exchangers, an incompressible fluid is heated initially and cooled at the boundary. The goal is to transfer the heat to the boundary as efficiently as possible. In this talk we study a related steady version of this problem: Consider the steady state temperature of in a fluid that is stirred, uniformly heated and cooled on the boundary. For a given large P\'eclet number, how should one stir to minimize the total heat? This problem was studied by Marcotte, Doering, Thiffeault and Young in '18, where the authors provided upper bounds using matched asymptotics. In this talk we will discuss how these upper bounds can be rigorously proved using large deviations, prove an almost matching lower bound in one case.
Joint work with Son Van (Carnegie Mellon University).
Friday, July 16, 11:30 ~ 11:55 UTC-3
Steady Rayleigh-Bénard convection between no-slip boundaries
Baole Wen
University of Michigan, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Buoyancy-driven flows are central to engineering heat transport, atmosphere and ocean dynamics, climate science, geodynamics, and stellar physics. Rayleigh-Bénard convection---the buoyancy driven flow in a fluid layer heated from below and cooled from above---is recognized as the simplest scenario in which to study such phenomena, and beyond its importance for applications this problem has served for a century as one of the primary paradigms of nonlinear physics, complex dynamics, pattern formation and turbulence. A central question about Rayleigh-Bénard convection is how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ and the Prandtl number $Pr$---i.e., how heat flux depends on imposed temperature gradient and the ratio of the fluid's kinematic viscosity to its thermal diffusivity---as $Ra\rightarrow\infty$. Experiments and simulations have yet to rule out either `classical' $Nu \sim Ra^{1/3}$ or `ultimate' $Nu \sim Ra^{1/2}$ asymptotic scaling. Here we provide clear quantitative evidence suggesting that the ultimate regime might not exist. Our tactic is to study relatively simple time-independent states called rolls and compare heat transport by these rolls with that of turbulent convection. These steady rolls are not typically seen in large-$Ra$ simulations or experiments because they are dynamically unstable. Nonetheless, they are part of the global attractor for the infinite-dimensional dynamical system defined by Rayleigh's model, and recent results suggest that steady rolls may be one of the key coherent states comprising the `backbone' of turbulent convection. By developing novel numerical methods, we compute steady rolls between no-slip boundaries for $Ra\le 10^{14}$ with $Pr=1$ and various horizontal periods. We find that rolls of the periods that maximize $Nu$ at each $Ra$ have classical $Nu\sim Ra^{1/3}$ scaling asymptotically, and they transport more heat than turbulent experiments or simulations at similar parameters. If turbulent heat transport continues to be dominated by steady transport asymptotically, it cannot achieve ultimate scaling.
Joint work with Charles R. Doering (University of Michigan, USA) and David Goluskin (University of Victoria, Canada).
Friday, July 16, 12:00 ~ 12:25 UTC-3
Optimal minimax bounds for the Navier-Stokes equations and other infinite dimensional dissipative systems
Ricardo Rosa
Universidade Federal do Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Obtaining sharp estimates for quantities involved in a given model is an integral part of a modeling process. For dynamical systems whose orbits display a complicated, perhaps chaotic, behaviour, the aim is usually to estimate time or ensemble averages of given quantities. This is the case, for instance, in turbulent flows. In this talk, the aim is to present a minimax optimization formula that yields optimal bounds for time and ensemble averages of dissipative infinite-dimensional systems, including the two- and three-dimensional Navier-Stokes equations. The results presented here are extensions to the infinite-dimensional setting of a recent result on the finite-dimensional case given by Tobasco, Goluskin, and Doering in 2017. The optimal result occurs in the form of a minimax optimization problem and does not require knowledge of the solutions, only the law of the system. The minimax optimization problem appears in the form of a maximization over a portion of the phase space of the system and a minimization over a family of auxiliary functions made of cylindrical test functionals defined on the phase space. The function to be optimized is the desired quantity plus the duality product between the law of the system and the derivative of the auxiliary function.
Joint work with Roger Temam (Indiana University).
Friday, July 16, 12:30 ~ 12:55 UTC-3
Global solution for some fluid dynamics models
Mouhamadou Sy
Imperial College London, UK - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the generalized SQG and the 3D Euler equations. We construct invariant probability measures on regular enough Sobolev spaces. We then deduce an almost sure global well-posedness result and long-time behavior properties.
Joint work with Juraj Foldes (University of Virginia).
Friday, July 16, 14:00 ~ 14:25 UTC-3
Flexibility, rigidity and stability of steady fluid motion.
Theodore Drivas
Stony Brook University, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We discuss the possibility and the impossibility of smooth deformations of equilibrium fluid states in the presence of symmetries. We will also make some remarks about the stability/instability of stationary states.
Joint work with Peter Constantin, Tarek Elgindi, Dan Ginsberg.
Friday, July 16, 14:30 ~ 14:55 UTC-3
A Bayesian Approach to Estimating Background Flows from a Passive Scalar
Justin Krometis
Virginia Tech, United States - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the Bayesian inverse problem of estimating a background flow field from the partial and noisy observation of a passive scalar (e.g., a solute concentration) governed by advection and diffusion. We provide conditions under which the inference is consistent, i.e., the posterior converges to a Dirac measure on the true flow as the number of observations grows large. We also attack the problem computationally by leveraging MCMC methods adapted in recent years to infinite-dimensional settings.
Joint work with Nathan Glatt-Holtz (Tulane University, United States) and Jeff Borggaard (Virginia Tech, United States).
Friday, July 16, 15:00 ~ 15:25 UTC-3
Solvable intermittent shell model of turbulence
Alexei A. Mailybaev
IMPA, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We introduce a shell model of turbulence featuring intermittent behaviour with anomalous power-law scaling of structure functions. This model is solved analytically with the explicit derivation of anomalous exponents. The solution associates the intermittency with the hidden symmetry for Kolmogorov multipliers, making our approach relevant for real turbulence.
Friday, July 16, 15:30 ~ 15:55 UTC-3
Similarity and multiscaling for extreme Reynolds pipe and channel flows
Fabio Ramos
Federal University of Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this work, we propose a physical phenomenology suggesting the existence of a transition in the scaling of the wall shear stress from the turbulent regime to the extreme Reynolds regime in channel and pipe flows. This transition leads to a power-law friction formula, which is ultimately associated to an incomplete similarity in the inner variables, and to a complete similarity in the outer variables for the mean velocity profile, which can be represented by a multiscale power-law formula.
Joint work with Gabriel Sanfins (UFRJ), Hamidreza Anbarlooei (UFRJ) and Daniel Cruz (UFRJ).
Monday, July 19, 16:00 ~ 16:25 UTC-3
Global existence for the 2D Kuramoto-Sivashinsky equation
Mazzucato Anna
Penn State University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation (KSE) in 2 space dimensions in the presence of growing modes. The KSE is a model of long-wave instability in dissipative systems.
Joint work with David Ambrose (Drexel).
Monday, July 19, 16:30 ~ 16:55 UTC-3
Traveling wave solutions to the free boundary Navier-Stokes equations
Ian Tice
Carnegie Mellon University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
Consider a layer of viscous incompressible fluid bounded below by a flat rigid boundary and above by a moving boundary. The fluid is subject to gravity, surface tension, and an external stress that is stationary when viewed in a coordinate system moving at a constant velocity parallel to the lower boundary. The latter can model, for instance, a tube blowing air on the fluid while translating across the surface. In this talk we will detail the construction of traveling wave solutions to this problem, which are themselves stationary in the same translating coordinate system. While such traveling wave solutions to the Euler equations are well-known, to the best of our knowledge this is the first construction of such solutions with viscosity.
Joint work with Giovanni Leoni (Carnegie Mellon University).
Monday, July 19, 17:00 ~ 17:25 UTC-3
Moffatt's Magnetic Relaxation Equations
Susan Friedlander
University of Southern California, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider a magnetic relaxation model ( MRE ) to describe a topology-preserving dissipative PDE whose solutions are conjectured to converge in the infinite time limit to an ideal magnetostatic equilibria B. As is well known, any such a vector field is also an example of an Euler equilibria for an ideal fluid. The MRE system is an active vector equation with a cubic nonlinearity and is thus unusual and challenging.
Joint work with Rajendra Beekie ( Courant Institute, USA ) and Vlad Vicol ( Courant Institute , USA).
Monday, July 19, 17:30 ~ 17:55 UTC-3
Electroconvection in fluids
Mihaela Ignatova
Temple University, US - This email address is being protected from spambots. You need JavaScript enabled to view it.
We describe results on an electroconvection model in fluids. The model consists of two dimensional Navier-Stokes equations driven by electrical and body forces, coupled to an advection and fractional diffusion equation for the surface charge density, driven by voltage applied at the boundary. We prove global regularity of solutions, and show that the long time behavior is described by a finite dimensional attractor. In the absence of body forces, the attractor reduces to a singleton, i.e. there is a unique, globally stable stationary solution.
Monday, July 19, 19:00 ~ 19:25 UTC-3
The limit Euler-$\alpha$ to Euler in domains with boundary
MILTON Lopes Filho
Universidade Federal do Rio de Janeiro, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
The Euler-$\alpha$ equations are a regularization of the incompressible Euler system, obtained by dynamical averaging of small scales. The problem of identifying the limiting behavior of solutions when $\alpha \to 0$ is similar to the analogous problem for vanishing viscosity, but with important differences. In this talk we discuss several results concerning this limit, assuming no-slip boundary conditions and initial data increasingly irregular.
Joint work with Adriana Valentina Busuioc (U. St. Etienne), Dragos Iftimie (U. Lyon) and Helena Nussenzveig Lopes (UFRJ).
Monday, July 19, 19:30 ~ 19:55 UTC-3
On some models of incompressible fluids containing small particles.
Gabriela Planas
Universidade Estadual de Campinas, Brazil - This email address is being protected from spambots. You need JavaScript enabled to view it.
We consider the $\alpha$-Navier-Stokes equations coupled with a Vlasov type equation to model the flow of an incompressible fluid containing small particles. We prove the existence of global weak solutions to the coupled system subject to periodic boundary conditions. Moreover, we investigate the regularity of weak solutions and the uniqueness of regular solutions. The convergence of its solutions to that of the Navier-Stokes-Vlasov equations when $ \alpha $ tends to zero is also established. Results are extended to the model with the diffusion of spray, i.e., to the $\alpha$-Navier-Stokes-Vlasov-Fokker-Planck equations.
Joint work with Cristyan Pinheiro (Universidade Estadual de Campinas, Brazil).
Monday, July 19, 20:00 ~ 20:25 UTC-3
On criticality of the Navier-Stokes diffusion
Zoran Grujić
University of Virginia, USA
It has been known since the work of J.-L. Lions in 1960s that the hyper-dissipative (HD) Navier-Stokes (NS) system is regular as long as the diffusion exponent beta is greater or equal to 5/4 (5/4 is critical in the sense that the unique scaling-invariance of the system takes place at the energy level).
The goal of this talk is to present a mathematical framework--based on the scale of sparseness of the super-level sets of the higher-order derivatives--in which a HD NS flow near a potential spatiotemporal singularity is classified in three categories: `turbulent' (higher-order derivatives are dominant), `steady' (derivatives of different orders are comparable), and `laminar' (lower-order derivatives are dominant). In the laminar scenario, the blow-up is ruled out for any beta greater or equal to one, with no assumptions. In the turbulent scenario, the blow-up is ruled out for any beta greater than one, with no assumptions. In the steady scenario, the blow-up is ruled out for any beta greater than one, with an assumption that the flow exhibits a certain `focusing' property in the vicinity of the potential singularity. This is a joint work with Liaosha Xu.
Joint work with Liaosha Xu (University of Virginia, USA).
Monday, July 19, 20:30 ~ 20:55 UTC-3
Vanishing viscosity and conserved quantities for 2D incompressible flow
Helena Nussenzveig Lopes
Universidade Federal do Rio de Janeiro, Brasil - This email address is being protected from spambots. You need JavaScript enabled to view it.
Weak solutions of the incompressible Euler equations which are weak limits of vanishing viscosity Navier-Stokes solutions inherit, in two dimensions, conservation properties which are not available for general weak solutions. In fluid domains with no boundary, research has focused on the behavior of kinetic energy and of the $p$-moments of vorticity. In this talk I will report on recent work in this direction, particularly the case of $p$-moments of vorticity, with $1 \leq p <\infty$.
Joint work with Christian Seis (Universitat Munster), Victor Navarro-Fernandez (Universitat Munster) and Emil Wiedemann (Universitat Ulm).