Workshop on Hyperbolic Conservation Laws and Infinite-Dimensional Dynamical Systems. Department of Mathematics, University of Pittsburgh

Transcription

1 Workshop on Hyperbolic Conservation Laws and Infinite-Dimensional Dynamical Systems Department of Mathematics, University of Pittsburgh March 30 - April 1, 2012 Program All talks are in Conference Room A on the 3rd floor of the University Club, 123 University Place, Pittsburgh, PA Sponsors: Mathematics Research Center (MRC), Department of Mathematics, and the Dietrich School of Arts and Sciences of the University of Pittsburgh. Organizers: Anna Vainchtein and Dehua Wang, on behalf of the Applied Analysis Group (Gunduz Caginalp, Ming Chen, Xinfu Chen, Huiqiang Jiang, Marta Lewicka, Bill Troy, Anna Vainchtein, and Dehua Wang).

2 Workshop on Hyperbolic Conservation Laws and Infinite-Dimensional Dynamical Systems University of Pittsburgh, March 30 - April 1, 2012 Schedule All talks are in Conference Room A on the 3rd floor of the University Club. Friday, 3/30 Saturday, 3/31 Sunday, 4/1 8:00-9:00am Breakfast 8:50-9:00am Welcome Breakfast Morning Session Chair: M. Slemrod Chair: G.-Q. Chen Chair: P. Bates 9:00-9:50am C. Dafermos P. Bates K. Trivisa 10:00-10:50am G.-Q. Chen N. J. Walkington Y. Zheng 10:50-11:20am Coffee break K 11:20am-12:10pm A. Vasseur N. Masmoudi M. Slemrod 12:10-2:00pm Lunch break The End Afternoon Session Chair: C. Dafermos Chair: F.-H. Lin 2:00-2:50pm F.-H. Lin A. Cherkaev 3:00-3:50pm T. Sideris A. Zarnescu 3:50-4:20pm Coffee break K 4:20-5:10pm A. Panchenko M. Weinstein Banquet, 6:30pm, Ballroom A 1st floor, University Club

3 3 Friday, March 30 8:00-8:50am: Continental breakfast 8:50-9:00am: Welcome and opening remarks Morning Session Chair: Marshall Slemrod 9:00-9:50: Constantine Dafermos, Brown University Maximal dissipation in equations of evolution 10:00-10:50: Gui-Qiang Chen, University of Oxford, UK Shock Reflection/Diffraction, Free Boundary Problems, and Nonlinear Conservation Laws of Mixed Type 10:50-11:20: Coffee Break K 11:20-12:10: Alexis Vasseur, University of Texas at Austin Relative entropy method applied to the stability of shocks for systems of conservation laws 12:10-2:00pm: Lunch Break Afternoon Session Chair: Constantine Dafermos 2:00-2:50: Fanghua Lin, New York University Topological vorticity and conserved geometric motion 3:00-3:50: Thomas Sideris, University of California at Santa Barbara Almost global existence of small solutions for 2D incompressible Hookean elastodynamics 3:50-4:20pm: Coffee Break K 4:20-5:10: Alexander Panchenko, Washington State University Mesoscopic continuum mechanics of large ODE systems

4 4 Saturday, March 31 8:00-9:00am: Continental breakfast Morning Session Chair: Gui-Qiang Chen 9:00-9:50: Peter Bates, Michigan State University The motion of particles driven by Allen-Cahn dynamics on the boundary of a smooth domain 10:00-10:50: Noel Walkington, Carnegie Mellon University Numerical approximation of the Ericksen Leslie equations 10:50-11:20: Coffee Break K 11:20-12:10: Nader Masmoudi, New York University Global existence of surface waves 12:10-2:00pm: Lunch Break Afternoon Session Chair: Fanghua Lin 2:00-2:50: Andrej Cherkaev, University of Utah Evolution of damageable structures 3:00-3:50: Arghir Zarnescu, University of Sussex, UK Thermodynamics, energetics and regularity for a De Gennes liquid crystal system 3:50-4:20pm: Coffee Break K 4:20-5:10: Michael Weinstein, Columbia University Localization and scattering of waves in microstructures 6:30pm: Banquet in Ballroom A 1st floor, University Club.

5 5 Sunday, April 1 8:00-9:00am: Continental breakfast Morning Session Chair: Peter Bates 9:00-9:50: Konstantina Trivisa, University of Maryland On kinetic models for the collective self-organization of agents 10:00-10:50: Yuxi Zheng, Yeshiva University Hyperbolic equations in liquid crystals 10:50-11:20: Coffee Break K 11:20-12:10: Marshall Slemrod, University of Wisconsin at Madison Admissibility of weak solutions for the compressible Euler equations, n 2 THE END.

6 6 Workshop on Hyperbolic Conservation Laws and Infinite-Dimensional Dynamical Systems University of Pittsburgh, March 30 - April 1, 2012 Abstracts Peter Bates, Michigan State University Title: The motion of particles driven by Allen-Cahn dynamics on the boundary of a smooth domain Abstract: We consider particles described as peak-like solutions to a singularly perturbed nonlinear parabolic partial differential equation. Minimal energy stationary states were shown to exist by Ni and Takagi in a series of papers where detailed qualitative properties of these states were also derived. Taking the gradient flow of the energy functional leads to a nonlinear parabolic equation and it is natural to ask about the motion of particles as dynamic peak-like solutions away from equilibrium. By proving an abstract theorem about the existence of a true invariant manifold in the neighborhood of an approximately invariant, approximately normally hyperbolic invariant manifold, we are able to answer this question, giving the global dynamics of a particle on the boundary of a smooth domain. Questions concerning the dynamics of particles in the interior of the domain or driven by Cahn-Hilliard or other evolution laws may possibly be addressed by this approach. Gui-Qiang Chen, University of Oxford, UK Title: Shock Reflection/Diffraction, Free Boundary Problems, and Nonlinear Conservation Laws of Mixed Type Abstract: Shock waves are steep fronts that propagate in the compressible fluids when the convective motion dominates the diffusion, which are fundamental in nature. When a shock wave hits an obstacle or a flying object meets a shock wave, shock reflection/diffraction phenomena occur. Mathematical treatments of shock reflection/diffraction naturally involve various free boundary problems and related nonlinear conservation laws of mixed type. In this talk we will focus first on the problem of shock reflection-diffraction by a wedge and describe how the global shock reflection-diffraction problem can be formulated as a free boundary problem for nonlinear conservation laws of mixed hyperbolic-elliptic type. Then we will discuss recent developments in attacking the shock reflection-diffraction problem, including the existence, stability, and regularity of global regular configurations. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, corner singularities, and optimal apriori estimates. Further recent developments on several related problems on shock reflection/diffraction and open problems in this direction will be also addressed. Based on joint work with M. Feldman, as well as M. Bae, X. Deng, and W. Xiang. Andrej Cherkaev, Title: Evolution of damageable structures University of Utah Abstract: The paper deals with dynamics and equilibrium states of mass-spring structures that change their morphology due to damage or breakage of springs. We consider

7 evolution paths of equilibrium states, variable sets of compatibility and equilibrium conditions, stochastic evolution of damage spread in lattices, and structural optimization. We also consider dynamics of damage, propagation of damage waves in chains and lattices. The study is applied to design of protective structures. Constantine Dafermos, Title: Maximal dissipation in equations of evolution Brown University Abstract: The lecture will discuss evolutionary equations, such as the Hunter-Saxton equation or hyperbolic conservation laws, with the property that the Cauchy problem admits a multitude of solutions along which a certain energy functional decays at different rates. The aim is to examine whether solutions that dissipate the energy at maximal rate command special status. Fanghua Lin, New York University Title: Topological vorticity and conserved geometric motion 7 Nader Masmoudi, Title: Global Existence of Surface Waves New York University Abstract: This is a joint work with P. Germain and J. Shatah. I will review some results about the global existence for gravity waves and for capillary waves. The main ingredient is the space-time resonance method. Alexander Panchenko, Washington State University Title: Mesoscopic continuum mechanics of large ODE systems Abstract: The main question addressed in the talk is how to obtain continuum equations for volume averages from the ODEs of classical particle dynamics. Balance equations for the average density, linear momentum, and energy were derived by Irving and Kirkwood, Noll, Hardy, Murdoch and others. The missing ingredient in these works was closure: the equations were exact but calculation of fluxes required solving the underlying ODE system. We present a closure approximation based on the use of regularized deconvolutions, and discuss the resulting constitutive equations. Examples include Lennard-Jones oscillator chains, granular acoustics and meshless discretizations of two-fluid flows. Thomas Sideris, University of California at Santa Barbara Title: Almost Global Existence of Small Solutions for 2D Incompressible Hookean Elastodynamics Abstract: We study the initial value problem for elasticity on all of R 2 in the incompressible Hookean case. If the initial displacement and velocity are of order ɛ in an appropriate energy norm, for sufficiently small ɛ, then there exists a smooth solution for a time interval of length greater than exp(c/ɛ), where C is independent of the initial data and ɛ. The proof combines energy estimates with a ghost weight, decay estimates in L 2 and L, and the null structure of the nonlinear terms. This is joint work with Zhen Lei and Yi Zhou of Fudan University.

8 8 Marshall Slemrod, University of Wisconsin at Madison Title: Admissibility of weak solutions for the compressible Euler equations, n 2 Abstract: This talk compares three popular notions of admissibility for weak solutions of the compressible isentropic Euler equations of gas dynamics: (i) the viscosity criterion, (ii) the entropy inequality (the thermodynamically admissible isentropic solutions),(iii) the viscosity-capillarity criterion. An exact summation of the Chapman-Enskog expansion for Grad s moment system suggests that it is the third criterion that is representing the kinetic theory of gases. This in turn may suggest that the cause of non-uniqueness for the weak solutions satisfying the second criterion is that the entropy inequality is not fully capturing information from kinetic theory. Konstantina Trivisa, University of Maryland Title: On kinetic models for the collective self-organization of agents Abstract: A class of kinetic flocking equations is analyzed and the global existence of weak solutions is established. The models under consideration include the kinetic Cucker-Smale equation with possibly non-symmetric flocking potential, the Cucker-Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor (2011). In the heart of the analysis are the velocity averaging lemma and the Schauder fixed point theorem. This is joint work with T. Karper and A. Mellet. Alexis Vasseur, University of Texas at Austin Title: Relative entropy method applied to the stability of shocks for systems of conservation laws Abstract: We develop a theory based on relative entropy to show stability and uniqueness of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of big amplitude), among bounded entropic weak solutions having an additional strong trace property. The existence of a convex entropy is needed. No BV estimate is needed on the weak solutions considered. The theory holds without smallness condition. The assumptions are quite general. For instance, the strict hyperbolicity is not needed globally. For fluid mechanics, the theory handles solutions with vacuum. Noel Walkington, Carnegie Mellon University Title: Numerical Approximation of the Ericksen Leslie Equations Abstract: The Ericksen Leslie equations model the motion of nematic liquid crytaline fluids. The equations comprise the linear and angular momentum equations with nonconvex constraints on the kinematic variables. These equations possess a Hamiltonian structure which reveals the subtle coupling of the two equations, and a delicate balance between inertia, transport, and dissipation. While a complete theory for the full nonlinear system is not yet available, many interesting sub-cases have been analyzed. This talk will focus on the development and analysis of numerical schemes which inherit the Hamiltonian structure, and hence stability, of the continuous problem. In certain situations compactness properties of the discrete solutions can be established which guarantee convergence of schemes.

9 Michael Weinstein, Columbia University Title: Localization and Scattering of Waves in Microstructures Abstract: We present results on i) an interesting low energy wave localization effect in microstructures (joint work with V. Duchene and I. Vukicevic); and ii) Dirac points, conical singularities in the dispersion surfaces associated with two-dimensional honeycomb periodic structures. These have received much recent attention due to their relevance to physical systems such as graphene. (joint work with C.L. Fefferman). Arghir Dani Zarnescu, University of Sussex, UK Title: Thermodynamics, energetics and regularity for a De Gennes liquid crystal system Abstract: We present a thermodynamically consistent model describing the time evolution of nematic liquid crystals in the framework of DeGennes Q-tensor theory. The thermal effects are present through the component of the free energy that accounts for intermolecular interactions. We identify the apriori estimates and construct global-in-time weak solutions for arbitrary physically relevant initial data. In the second part of the talk, and for the case when the temperature is ignored, we prove global regularity in 2D and estimates on the rate of increase of the high norms. The first part is joint work with E. Feireisl, E. Rocca and G. Schimperna, while the second part is joint work with M. Paicu. Yuxi Zheng, Yeshiva University Title: Hyperbolic equations in liquid crystals Abstract: We study a class of variational wave systems modeling nematic liquid crystals. We establish singularity formation, existence, stability and semi-group properties of global weak solutions to the initial value problem. Based on joint work with A. Bressan, R. Glassey, J. Hunter, and Ping Zhang. 9