Workshop onPDEs in Fluid Dynamics

Department of Mathematics, University of Pittsburgh

November 3-5, 2017

Program

All talks are in Thackerary Hall 704 in the Department of Mathematics, Pittsburgh, PA15260.

Sponsors: Mathematics Research Center (MRC) of the University of Pittsburgh.

Organizers: Ming Chen and Dehua Wang.

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Friday, November 3

v Afternoon Session Chair: Dehua Wang

• 3:30-4:20: Eduard Feireisl, Czech Academy of Sciences(joint with department colloquium)

Weak and measure-valued solutions to the full Euler system

• 4:30-5:20: Changyou Wang, Purdue UniversityOn static and hydrodynamic theory of biaxial nematics

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Saturday, November 4

v Morning Session Chair: Gautam Iyer

• 8:30-9:20: Cheng Yu, University of Texas at AustinEnergy conservation for inhomogeneous Euler equations

• 9:30-10:20: Vlad Vicol, University of MinnesotaNonuniqueness of weak solutions to the Navier-Stokes equations

10:20-11:00: Coffee Break K

• 11:00-11:50: Eitan Tadmor, University of MarylandRegularity and emergence of flocking in PDE models with a commutatorstructure

12:00-2:00pm: Lunch Break

v Afternoon Session Chair: Ming Chen

• 2:00-2:50: Yan Guo, Brown UniversityL6 estimate for steady Boltzmann and its Navier Stokes’ limit

• 3:00-3:50: Ian Tice, Carnegie Mellon UniversityThe stability of contact lines in fluids

3:50-4:30pm: Coffee Break K

• 4:30-5:20: Wei Xiang, City University of Hong KongCompactness on Multidimensional Steady Euler Equations

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Sunday, November 5

v Morning Session Chair: Ian Tice

• 8:30-9:20: Mikhail Feldman, University of Wisconsin at MadisonUniqueness for shock reflection problem

• 9:30-10:20: Gautam Iyer, Carnegie Mellon UniversityAnomalous diffusion in passive scalar transport

10:20-11:00: Coffee Break K

• 11:00-11:50: Runzhang Xu, Harbin Engineering UniversityGlobal existence and blowup of solutions for the multidimensional sixth order“good” Boussinesq equation

12:00-2:00pm: Lunch Break

v Afternoon Session Chair: Huiqiang Jiang

• 2:00-2:50: Feng Xie, Shanghai Jiaotong UniversityPrandtl boundary layer expansion analysis for MHD equations

• 3:00-3:50: Free discussion

THE END.

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Workshop on PDEs in Fluid Dynamics

University of Pittsburgh, November 3-5, 2017

Abstracts

Eduard Feireisl, Czech Academy of Sciences

Title: Weak and measure-valued solutions to the full Euler system

Abstract: We introduce a concept of dissipative measure valued solution for the full Eulersystem describing the motion of an inviscid fluid. We show the existence of non-trivialsolutions of this type by proving the existence of infinitely many weak (distributional)solutions to the same system. Then we show several applications in problems of singularlimits where either the primitive or target system is considered in the measure-valuedsense.

Mikhail Feldman, University of Wisconsin at Madison

Title: Uniqueness for shock reflection problem

Abstract: We discuss shock reflection problem for compressible gas dynamics, von Neu-mann conjectures on transition between regular and Mach reflections, and existence ofregular reflection solutions for potential flow equation. Then we will talk about recent re-sults on uniqueness of regular reflection solutions for potential flow equation in a naturalclass of self-similar solutions. The approach is to reduce the shock reflection problem to afree boundary problem for a nonlinear elliptic equation, and prove uniqueness by a versionof method of continuity. A property of solutions important for the proof of uniqueness isconvexity of the free boundary. This talk is based on joint work with G.-Q. Chen and W.Xiang.

Yan Guo, Brown University

Title: L6 estimate for steady Boltzmann and its Navier Stokes’ limit

Abstract: We present a new L6 estimate in the derivation of steady Navier-Stokes equa-tions from the Boltzmann theory.

Gautam Iyer, Carnegie Mellon University

Title: Anomalous diffusion in passive scalar transport

Abstract: Consider a diffusive passive scalar advected by a two dimensional incom-pressible flow. If the flow is cellular (i.e. has a periodic Hamiltonian with no unboundedtrajectories), then classical homogenization results show that the long time behaviour isan effective Brownian motion. We show that on intermediate time scales, the effectivebehaviour is instead a fractional kinetic process. At the PDE level this means that whilethe long time scaling limit is the heat equation, the intermediate time scaling limit is atime fractional heat equation. We will also describe the expected intermediate behaviourin the presence of open channels.

Time permitting, in the last part of the talk we will describe a few other trap modelsthat arise in PDE homogenization limits that exhibit a similar behaviour on intermediatetime scales.

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Eitan Tadmor, University of Maryland

Title: Regularity and emergence of flocking in PDE models with a commutator structure

Abstract: We discuss the global regularity for a general class of Eulerian dynamics drivenby a forcing with a commutator structure. The study of such systems is motivated by thehydrodynamic description of agent-based models for flocking driven by alignment. Forcommutators involving bounded kernels, existence of strong solutions follows for initialdata which are sub-critical, namely – the initial divergence is “not too negative and theinitial spectral gap is “not too large. Singular kernels, corresponding to fractional Laplacianof order 0 < s < 1, behave better: global regularity persists and flocking follows. Singu-larity helps!. A similar role of the spectral gap is found in our study of two-dimensionalpressure-less equations, corresponding to the formal limit s = 0, proving the existence ofweak dual solutions as vanishing viscosity limits.

Ian Tice, Carnegie Mellon University

Title: The stability of contact lines in fluids

Abstract: The contact line problem in interfacial fluid mechanics concerns the triple-junction between a fluid, a solid, and a vapor phase. Although the equilibrium configu-rations of contact lines have been well-understood since the work of Young, Laplace, andGauss, the understanding of contact line dynamics remains incomplete and is a source ofwork in experimentation, modeling, and mathematical analysis. In this talk we considera 2D model of contact point (the 2D analog of a contact line) dynamics for an incom-pressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. Themodel allows for fully dynamic contact angles and points. We show that small perturba-tions of the equilibrium configuration give rise to global-in-time solutions that decay toequilibrium exponentially fast. This is joint with with Yan Guo.

Vlad Vicol, University of Minnesota

Title: Nonuniqueness of weak solutions to the Navier-Stokes equations

Abstract: We prove that weak solutions of the Navier-Stokes equations are not uniquein the class of weak/mild solutions with finite kinetic energy. Further results will bediscussed. This is joint work with Tristan Buckmaster.

Changyou Wang, Purdue University

Title: On static and hydrodynamic theory of biaxial nematics

Abstract: In this talk, I will describe a simplified form of the Landau-De Gennes Q-tensor theory on biaxial nematic liquid crystals, proposed by Govers, Vertogen, and Leslieback in 1980’s. I will then present some analytic results on both the equilibrium equationand its (hydro)dynamic equation.

Wei Xiang, City University of Hongkong

Title: Compactness on Multidimensional Steady Euler Equations

Abstract: In this talk, we will introduce the compactness frame work for approximatesolutions to the incompressible limits governed by the steady compressible full Euler equa-tions in arbitrary dimension. Then we will show the latest progress on the Euler flow with

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contact discontinuities in in finitely long nozzles, which relies on the above compactnessframework. These are the joint works with G.-Q. Chen, F.-M. Huang and T.-Y. Wang.

Feng Xie, Shanghai Jiaotong University

Title: Prandtl boundary layer expansion analysis for MHD equations

Abstract: In this talk, I first introduce some mathematical results and methods in thestudy of classical Prandtl boundary layer theory. Then, I will focus on the related well-posedness and convergence theories for the characteristic MHD boundary layer. Under theassumption that the initial tangential magnetic field is not zero, we establish the local-in-time existence and uniqueness of solution to the nonlinear MHD boundary layer equations.Moreover, based on the multi-scale expansions, we justify the vanishing viscosity andmagnetic diffusion limit process in L∞ sense by weighted energy estimates in Sobolevspaces. This justifies the physical understanding that the magnetic field has a stabilizingeffect on MHD boundary layer in rigorous mathematics.

Runzhuang Xu, Harbin Engineering University

Title: Global existence and blowup of solutions for the multidimensional sixth-order“good Boussinesq equation

Abstract: This talk is concerned with the Cauchy problem of solutions for some nonlinearmultidimensional “good Boussinesq equation of sixth order at three different initial energylevels. In the framework of potential well, the global existence and blowup of solutions areobtained together with the concavity method at both low and critical initial energy level.Moreover by introducing a new stable set, we present some sufficient conditions on initialdata such that the weak solution exists globally at supercritical initial energy level.

Cheng Yu, University of Texas at Austin

Title: Energy conservation for inhomogeneous Euler equations

Abstract: In this joint work with Ming Chen, we consider the problem of energy conser-vation for the two- and three-dimensional density-dependent Euler equations. Two typesof sufficient conditions on the regularity of solutions are provided to ensure the conser-vation of total kinetic energy on the entire time interval including the initial time. Thefirst class of data assumes integrability on the spatial gradient of the density, and hencecovers the classical result of Constantin-E-Titi for the homogeneous Euler equations. Theother type of data imposes extra time Besov regularity on the velocity profile, and thecorresponding result can be applied to deal with a wide class of rough density profiles.