Workshop on PDEs in Fluid Dynamics. Department of Mathematics, University of Pittsburgh. November 3-5, Program

Transcription

1 Workshop on PDEs 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, PA Sponsors: Mathematics Research Center (MRC) of the University of Pittsburgh. Organizers: Ming Chen and Dehua Wang. 1

2 2 Friday, November 3 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 University On static and hydrodynamic theory of biaxial nematics

3 3 Saturday, November 4 Morning Session Chair: Gautam Iyer 8:30-9:20: Cheng Yu, University of Texas at Austin Energy conservation for inhomogeneous Euler equations 9:30-10:20: Vlad Vicol, University of Minnesota Nonuniqueness of weak solutions to the Navier-Stokes equations 10:20-11:00: Coffee Break 11:00-11:50: Eitan Tadmor, University of Maryland Regularity and emergence of flocking in PDE models with a commutator structure 12:00-2:00pm: Lunch Break Afternoon Session Chair: Ming Chen 2:00-2:50: Yan Guo, Brown University L 6 estimate for steady Boltzmann and its Navier Stokes limit 3:00-3:50: Ian Tice, Carnegie Mellon University The stability of contact lines in fluids 3:50-4:30pm: Coffee Break 4:30-5:20: Wei Xiang, City University of Hong Kong Compactness on Multidimensional Steady Euler Equations

4 4 Sunday, November 5 Morning Session Chair: Ian Tice 8:30-9:20: Mikhail Feldman, University of Wisconsin at Madison Uniqueness for shock reflection problem 9:30-10:20: Gautam Iyer, Carnegie Mellon University Anomalous diffusion in passive scalar transport 10:20-11:00: Coffee Break 11:00-11:50: Runzhang Xu, Harbin Engineering University Global existence and blowup of solutions for the multidimensional sixth order good Boussinesq equation 12:00-2:00pm: Lunch Break Afternoon Session Chair: Huiqiang Jiang 2:00-2:50: Feng Xie, Shanghai Jiaotong University Prandtl boundary layer expansion analysis for MHD equations 3:00-3:50: Free discussion THE END.

5 5 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 Euler system describing the motion of an inviscid fluid. We show the existence of non-trivial solutions 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 singular limits where either the primitive or target system is considered in the measure-valued sense. Mikhail Feldman, University of Wisconsin at Madison Title: Uniqueness for shock reflection problem Abstract: We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary. This talk is based on joint work with G.-Q. Chen and W. Xiang. Yan Guo, Brown University Title: L 6 estimate for steady Boltzmann and its Navier Stokes limit Abstract: We present a new L 6 estimate in the derivation of steady Navier-Stokes equations 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 incompressible flow. If the flow is cellular (i.e. has a periodic Hamiltonian with no unbounded trajectories), then classical homogenization results show that the long time behaviour is an effective Brownian motion. We show that on intermediate time scales, the effective behaviour is instead a fractional kinetic process. At the PDE level this means that while the long time scaling limit is the heat equation, the intermediate time scaling limit is a time fractional heat equation. We will also describe the expected intermediate behaviour in the presence of open channels. Time permitting, in the last part of the talk we will describe a few other trap models that arise in PDE homogenization limits that exhibit a similar behaviour on intermediate time scales.

6 6 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 driven by a forcing with a commutator structure. The study of such systems is motivated by the hydrodynamic description of agent-based models for flocking driven by alignment. For commutators involving bounded kernels, existence of strong solutions follows for initial data which are sub-critical, namely the initial divergence is not too negative and the initial spectral gap is not too large. Singular kernels, corresponding to fractional Laplacian of order 0 < s < 1, behave better: global regularity persists and flocking follows. Singularity helps!. A similar role of the spectral gap is found in our study of two-dimensional pressure-less equations, corresponding to the formal limit s = 0, proving the existence of weak 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 triplejunction between a fluid, a solid, and a vapor phase. Although the equilibrium configurations of contact lines have been well-understood since the work of Young, Laplace, and Gauss, the understanding of contact line dynamics remains incomplete and is a source of work in experimentation, modeling, and mathematical analysis. In this talk we consider a 2D model of contact point (the 2D analog of a contact line) dynamics for an incompressible, viscous, Stokes fluid evolving in an open-top vessel in a gravitational field. The model allows for fully dynamic contact angles and points. We show that small perturbations of the equilibrium configuration give rise to global-in-time solutions that decay to equilibrium 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 unique in the class of weak/mild solutions with finite kinetic energy. Further results will be discussed. 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 Leslie back in 1980 s. I will then present some analytic results on both the equilibrium equation and 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 approximate solutions to the incompressible limits governed by the steady compressible full Euler equations in arbitrary dimension. Then we will show the latest progress on the Euler flow with

7 contact discontinuities in in finitely long nozzles, which relies on the above compactness framework. 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 the study of classical Prandtl boundary layer theory. Then, I will focus on the related wellposedness and convergence theories for the characteristic MHD boundary layer. Under the assumption that the initial tangential magnetic field is not zero, we establish the local-intime existence and uniqueness of solution to the nonlinear MHD boundary layer equations. Moreover, based on the multi-scale expansions, we justify the vanishing viscosity and magnetic diffusion limit process in L sense by weighted energy estimates in Sobolev spaces. This justifies the physical understanding that the magnetic field has a stabilizing effect 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 nonlinear multidimensional good Boussinesq equation of sixth order at three different initial energy levels. In the framework of potential well, the global existence and blowup of solutions are obtained 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 initial data 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 conservation for the two- and three-dimensional density-dependent Euler equations. Two types of sufficient conditions on the regularity of solutions are provided to ensure the conservation of total kinetic energy on the entire time interval including the initial time. The first class of data assumes integrability on the spatial gradient of the density, and hence covers the classical result of Constantin-E-Titi for the homogeneous Euler equations. The other type of data imposes extra time Besov regularity on the velocity profile, and the corresponding result can be applied to deal with a wide class of rough density profiles. 7