VIRTUAL ONLY: Advances and Challenges in Hyperbolic Conservation Laws
Institute for Computational and Experimental Research in Mathematics (ICERM)
May 17, 2021  May 21, 2021
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Monday, May 17, 2021

9:45  10:00 am EDTWelcomeVirtual
 Brendan Hassett, ICERM/Brown University

10:00  10:30 am EDTExistence theory for viscoelasticity of KelvinVoigt type with nonconvex stored energiesVirtual
 Speaker
 Athanasios Tzavaras, King Abdullah University of Science and Technology
 Session Chair
 Alberto Bressan, The Pennsylvania State University
Abstract
I will review the existence and uniqueness theory for viscoelasticity of KelvinVoigt type with nonconvex stored energies. The analysis is based on propagation of $H^1$regularity for the deformation gradient of weak solutions in two and three dimensions assuming that the stored energy satisfies the AndrewsBall condition, in particular allowing for nonmonotone stresses. By contrast, a counterexample indicates that for nonmonotone stressstrain relations (even in 1d) initial oscillations of the strain lead to solutions with sustained oscllations. In two space dimensions, it turns out that weak solutions with deformation gradient in $H^1$ are in fact unique, providing a striking analogy to corresponding results in the theory of 2D Euler equations with bounded vorticity.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:30 am EDTHard spheres dynamics: weak vs strong collisionsVirtual
 Speaker
 Denis Serre, Ecole Normale Supérieure de Lyon
 Session Chair
 Alberto Bressan, The Pennsylvania State University
Abstract
Hard spheres dynamics is among the finest ones in the scale of models for gas dynamics. Before passing to the limit in order to derive a kinetic model, one needs to understand how many collisions take place, and how strong of weak they are. Collisions can be exponentially many in terms of the number $N$ of particles, but we prove that at most $O(N^2)$ among them can be significant.

11:45 am  1:30 pm EDTLunch/Free TimeVirtual

1:30  2:00 pm EDTThe Future of Front TrackingVirtual
 Speaker
 James Glimm, Stony Brook University
 Session Chair
 GuiQiang Chen, University of Oxford
Abstract
Solutions of the NavierStokes equation are not unique. We show that a maximum rate of entropy production is necessary admissibility condition to achieve a physical solution. For incompressible, constant density flow, the resulting solution is probably unique. For variable density flows, multiple distinct flow regimes are possible, and are determined by new physics added at the microscale. Front tracking has a role to enforce some examples of microphysical turbulent mixing regimes. A new multidimensional random choice method appears to be the ideal choice among competing sharp interface algorithms for this purpose.

2:15  2:30 pm EDTBreakCoffee Break  Virtual

2:30  3:00 pm EDTStructure preserving numerical methods for hyperbolic conservation and balance lawsVirtual
 Speaker
 Alina Chertock, North Carolina State University
 Session Chair
 GuiQiang Chen, University of Oxford
Abstract
Any physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable highresolution numerical methods still remains to be an extremely challenging task. In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) wellbalanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steadystate solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest.

3:15  3:45 pm EDTEulerian Lagrangian discontinuous Galerkin methods for nonlinear transport problemsVirtual
 Speaker
 Jingmei Qiu, University of Delaware
 Session Chair
 GuiQiang Chen, University of Oxford
Abstract
We propose a new EulerianLagrangian (EL) discontinuous Galerkin (DG) method formulated by introducing a modified adjoint problem for the test function and by performing the integration of PDE over a spacetime region partitioned by timedependent linear functions approximating characteristics. The error incurred in characteristics approximation in the modified adjoint problem can then be taken into account by a new flux term, and can be integrated by methodofline RungeKutta (RK) methods. The ELDG framework is designed as a generalization of the semiLagrangian (SL) DG method and classical Eulerian RK DG method for linear advection problems. It takes advantages of both formulations. In the EL DG framework, characteristics are approximated by a linear function in time, thus shapes of upstream cells are quadrilaterals in general twodimensional problems. No quadraticcurved quadrilaterals are needed to design higher than second order schemes as in the SL DG scheme. On the other hand, the time step constraint from a classical Eulerian RK DG method is greatly mitigated, as it is evident from our theoretical and numerical investigations. Connection of the proposed EL DG method with the arbitrary LagrangianEulerian (ALE) DG is observed. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

4:00  5:00 pm EDTGathertown ReceptionReception  Virtual
Tuesday, May 18, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am EDTShock Waves and EntropyVirtual
 Speaker
 TaiPing Liu, Stanford University
 Session Chair
 Constantine Dafermos, Brown University
Abstract
It was recognized soon after the pioneering works of Riemann and Stokes in the midnineteenth century that entropy increases as the gas flows across a shock wave for polyatomic gases. Around 1940's Bethe and Weyl independently formulated the convexity condition for the equivalence of the compressibility of a shock and the entropy increase across it. This was subsequently generalized to the general system of hyperbolic conservation laws by Lax. The situation without convexity is interesting. The Russian school of Oleinik and Krushkov obtained complete results for scalar laws. It is understood now that the existence of entropy for a system is a constitutive hypothesis. Godunov established the relation between the existence of entropy and the symmetric structure of a system. There have been efforts to relate the admissibility conditions for shock waves to the entropy production. For this, we offer a definitive result for shock waves in the Euler equations for compressible media. In this talk, we will survey the historical developments on general systems as well as some exact analysis for the Euler equations.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:30 am EDTBV Solutions to a Hydrodynamic Limit of Flocking TypeVirtual
 Speaker
 Cleopatra Christoforou, University of Cyprus
 Session Chair
 Constantine Dafermos, Brown University
Abstract
Mathematical models introduced to capture the emergent behavior of selforganized systems have brought new challenges in the mathematical community and a lot of attention in the recent years. Most studies on flocking models have been on the behavior of the particle model or the corresponding kinetic formulation and its hydrodynamic formulation that provides in the limit an Eulertype flocking system. This area has been investigated so far in the context of smooth solutions. In this talk, we will discuss a hydrodynamic model of flocking type in the setting of entropy weak solutions. We (i) establish global existence of entropy weak solutions for arbitrary initial data of bounded variation with finite mass confined in a bounded interval and uniformly positive density therein and (ii) show that the entropy solution admits time asymptotic flocking. This is a joint work with Debora Amadori from University of L’Aquila.

11:45 am  1:30 pm EDTLunch/Free TimeVirtual

1:30  2:00 pm EDTLocally dissipative solutions of the Euler equationsVirtual
 Speaker
 Camillo De Lellis, IAS
 Session Chair
 Fengyan Li, Rensselaer Polytechinic Institute
Abstract
The Onsager conjecture, recently solved by Phil Isett, states that, below a certain threshold regularity, Hoelder continuous solutions of the Euler equations might dissipate the kinetic energy. The original work of Onsager was motivated by the phenomenon of anomalous dissipation and a rigorous mathematical justification of the latter should show that the energy dissipation in the NavierStokes equations is, in a suitable statistical sense, independent of the viscosity. In particular it makes much more sense to look for solutions of the Euler equations which, besides dissipating the {\em total} kinetic energy, satisfy as well a suitable form of local energy inequality. Such solutions were first shown to exist by Laszlo Szekelyhidi Jr. and myself. In this talk I will review the methods used so far to approach their existence and the most recent results by Isett and by Hyunju Kwon and myself.

2:15  2:30 pm EDTBreakCoffee Break  Virtual

2:30  3:00 pm EDTShock reflection problems: existence, stability and regularity of global solutionsVirtual
 Speaker
 Mikhail Feldman, University of Wisconsin
 Session Chair
 Fengyan Li, Rensselaer Polytechinic Institute
Abstract
In this talk we will start with discussion of shock reflection phenomena, and von Neumann conjectures on transition between regular and Mach reflections. Then we describe the results on existence, uniqueness, stability and regularity of global solutions to shock reflection for potential flow, and discuss the techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation in selfsimilar coordinates, with ellipticity degenerate near a part of the boundary. We will also discuss the open problems in the area.

3:15  3:45 pm EDTCan the flow behind a weakshock Mach reflection be shockfree?Virtual
 Speaker
 John Hunter, University of California Davis
 Session Chair
 Fengyan Li, Rensselaer Polytechinic Institute
Abstract
Numerical solutions of weakshock Mach reflections show a sequence of supersonic patches and triple points in a tiny region below the leading triple point. A basic question, with analogs to the existence of shockfree flows over transonic airfoils, is whether it is possible to have only a single shockfree supersonic patch and triple point behind the Mach stem, or must there be multiple triple points. We explore this question using the steady transonic small disturbance equation as the simplest model equation. Assuming the hodograph transformation is invertible near the triple point, we formulate an oblique derivative Tricomi problem for the Tricomi equation as a local description of shockfree flows behind the Mach stem and discuss its solvability.
Wednesday, May 19, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am EDTAsymptotical stability of wave patterns for viscous conservation laws under periodic perturbationsVirtual
 Speaker
 Feimin Huang, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
 Session Chair
 ChiWang Shu, Brown University
Abstract
The asymptotical stability of wave patterns for viscous conservation laws is an interesting and important problem. Considerable progress on the stability of single shock wave, rarefaction wave in the whole space has been achieved in the past several decades. An interesting question is whether these wave patterns are still stable under periodic spatial perturbation. Compared with the stability problem in the whole space, the solution has no limit at the far fields. In this talk, I will present recent works on the stability of single shock wave and rarefaction wave for viscous conservation laws under periodic perturbations.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:30 am EDTObstacle problem, Euler system and turbulenceVirtual
 Speaker
 Eduard Feireisl, Czech Academy of Sciences
 Session Chair
 ChiWang Shu, Brown University
Abstract
We consider a statistical limit of solutions to the compressible NavierStokes system in the high Reynolds number regime in a domain exterior to a rigid body. We investigate to what extent this highly turbulent regime can be modeled by an external stochastic perturbation, as suggested in the related physics literature.To this end, we interpret the statistical limit as a stochastic process on the associated trajectory space. We suppose that the limit process is statistically equivalent to a solution of the stochastic compressible Euler system.Then, necessarily,
1. the stochastic forcing is not active  the limit is a statistical solution of the deterministic Euler system;
2. the solutions Sconverge to the limit;
3. if, in addition, the expected value of the limit process solves the Euler system, then the limit is deterministic and the convergence is strong a.a.
These results strongly indicate that a stochastic forcing may not be a suitable model for turbulent randomness in compressible fluid flows. 
11:45 am  1:30 pm EDTLunch/Free TimeVirtual

1:30  2:30 pm EDTPoster SessionVirtual

2:30  3:00 pm EDTDevising energyconserving finite element methods for wave propagationVirtual
 Speaker
 Bernardo Cockburn, University of Minnesota
 Session Chair
 Eitan Tadmor, University of Maryland
Abstract
Discontinuous Galerkin methods for the wave equation are typically defined by using the fact that the equation can be expressed as a symmetric hyperbolic system. Although these methods can be devised to be highorder accurate, they are naturally dissipative and cannot be used for longtime simulations. We show that it is possible, by taking advantage of the Hamiltonian structure of the wave equation, to overcome this drawback and obtain Discontinuous Galerkin (and other finite element) methods which maintain their original highorder accuracy while conserving the discrete spacediscretization energy. We sketch the extension of this approach to other systems of equations with Hamiltonian structure including equations modeling linear and nonlinear elastic, electromagnetic and water wave equations.

3:15  3:30 pm EDTBreakCoffee Break  Virtual

3:30  4:00 pm EDTHigh Order Unconditionally Strong Stability Preserving MultiDerivative Implicit and Imex Runge–Kutta Methods with Asymptotic Preserving PropertiesVirtual
 Speaker
 Sigal Gottlieb, University of Massachusetts Dartmouth
 Session Chair
 Eitan Tadmor, University of Maryland
Abstract
In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit multiderivative Runge–Kutta schemes, and SSP implicitexplicit (IMEX) multiderivative Runge–Kutta schemes where the timestep restriction is independent of the stiff term. The unconditional SSP property for a method of order p > 2 is unique among SSP methods, and depends on a backwardintime assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge–Kutta schemes of order up to p = 4, and IMEX Runge–Kutta schemes of order up to p = 3. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the BhatnagarGrossKrook (BGK) kinetic equation. We present numerical results to support the theoretical results, on a variety of problems.
Thursday, May 20, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am EDTNumerical methods for conservation laws on graphsVirtual
 Speaker
 Ulrik Fjordholm, University of Oslo
 Session Chair
 Konstantina Trivisa, University of Maryland
Abstract
We consider a set of scalar conservation laws on a graph. Based on a choice of stationary states of the problem – analogous to the constants in Kruzkhov's entropy condition – we establish the uniqueness and stability of entropy solutions. For two classes of flux functions – either monotone or concave fluxes – we establish the convergence of an easytoimplement Engquist–Oshertype finite volume method. This is joint work with Markus Musch and Nils Henrik Risebro (University of Oslo).

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:30 am EDTThe nonlinear stability of Schwarzschild black holes in general relativityVirtual
 Speaker
 Mihalis Dafermos, Princeton University
 Session Chair
 Konstantina Trivisa, University of Maryland

11:45 am  1:30 pm EDTLunch/Free TimeVirtual

1:30  2:00 pm EDTVariations on the theme of scalar conservation lawsVirtual
 Speaker
 Wen Shen, The Pennsylvania State University
 Session Chair
 Dehua Wang, University of Pittsburgh
Abstract
In this talk we visit several variations of a standard conservation law. These include the cases such as: (i) the flux function is discontinuous in space and time, (ii) the flux function is nonlocal and contains an integral term, and (iii) combinations of them. We discuss key challenges, some recent results, and applications to traffic flow.

2:15  2:30 pm EDTBreakCoffee Break  Virtual

2:30  3:00 pm EDTOn the Dynamics of Ferrofluids: Weak Solutions and Relaxation Limit for the Rosensweig ModelVirtual
 Speaker
 Franziska Weber, Carnegie Mellon University
 Session Chair
 Dehua Wang, University of Pittsburgh
Abstract
We show existence of global weak solutions of the Rosensweig model of ferrofluids, using DiPernaLions’ theory of compressible fluids. Then, we investigate the relaxation to equilibrium ϵ→0 using the relative entropy method. If the limiting system has a Lipschitz continuous solution, we can show a convergence rate in ϵ, if the limiting system has only a weak solution, we obtain strong convergence of a subsequence in L^2.

3:15  3:45 pm EDTArbitrary order structure preserving discontinuous Galerkin methods for hyperbolic balance lawsVirtual
 Speaker
 Yulong Xing, The Ohio State University
 Session Chair
 Dehua Wang, University of Pittsburgh
Abstract
Euler equations under gravitational fields and shallow water equations with a nonflat bottom topography are two prototype hyperbolic conservation laws with source term (also referred as hyperbolic balance laws). They both have various applications in many fields. In this presentation, we will talk about arbitrary order structure preserving discontinuous Galerkin finite element methods which can exactly capture the nontrivial steady state solutions of these models, and at the same time maintain the nonnegativity of some physical quantities. Numerical tests are provided to verify the wellbalanced property, positivitypreserving property, highorder accuracy, and good resolution for both smooth and discontinuous solutions.
Friday, May 21, 2021

9:00  9:45 am EDTGathertown Morning CoffeeCoffee Break  Virtual

10:00  10:30 am EDTOn the singular local limit for conservation laws with nonlocal fluxesVirtual
 Speaker
 Laura Spinolo, IMATICNR
 Session Chair
 Mikhail Feldman, University of Wisconsin
Abstract
Conservation laws with nonlocal fluxes have recently drawn considerable attention owing to their applications to several engineering problems, like models of vehicular and pedestrian traffic. They consist of conservation laws where the flux function depends on the convolution of the solution with a given kernel. In the singular local limit where the convolution kernel is replaced by a Dirac delta, one formally recovers a (classical) conservation law. In this talk I will overview recent progress on the rigorous justification of this nonlocaltolocal limit. I will mention counterexamples showing that, in general, the solutions of the nonlocal problems do not converge to the entropy admissible solution of the conservation law. On the other hand, the nonlocaltolocal limit have been recently justified, under different assumptions, in the case of anisotropic convolution kernels, which are natural in view of applications to models of vehicular traffic. The talk will be based on joint works with Maria Colombo, Gianluca Crippa and Elio Marconi.

10:45  11:00 am EDTBreakCoffee Break  Virtual

11:00  11:30 am EDTFast computations of high order WENO methods for hyperbolic conservation lawsVirtual
 Speaker
 Yongtao Zhang, University of Notre Dame
 Session Chair
 Mikhail Feldman, University of Wisconsin
Abstract
Weighted essentially nonoscillatory (WENO) schemes are a popular class of high order accurate numerical methods for solving hyperbolic conservation laws. They have been applied extensively in computational fluid dynamics and other scientific problems. However, for complicated multidimensional problems, it often leads to large amount of operations and computational costs in the numerical simulations by using nonlinear high order accuracy WENO schemes such as fifth order WENO schemes. How to achieve fast simulations by high order WENO methods for solving hyperbolic conservation laws is a challenging and important question. In this talk, I shall present our recent work on applying fast sweeping methods and sparsegrid techniques for efficient computations of WENO schemes. Fast sweeping methods are a class of efficient iterative methods for solving steady state problems of hyperbolic PDEs. They utilize alternating sweeping strategy to cover a family of characteristics in a certain direction simultaneously in each sweeping order. Coupled with the GaussSeidel iterations, these methods can achieve a fast convergence speed for computations of steady state solutions of hyperbolic PDEs. We design absolutely convergent fixedpoint fast sweeping WENO methods for solving steady state solutions of hyperbolic conservation laws. For highdimensional problems, sparsegrid techniques are efficient approximation tools to reduce degrees of freedom in the discretizations. We apply the sparsegrid combination technique to fifth order WENO finite difference schemes for solving time dependent hyperbolic PDEs defined on high spatial dimension domains. Extensive numerical experiments shall be shown to demonstrate large savings of computational costs by comparing with simulations using traditional methods for solving hyperbolic conservation laws.

11:45 am  1:15 pm EDTLunch/Free TimeVirtual

1:15  1:45 pm EDTCoupled parabolichyperbolic moving boundary problems in poroelasticityVirtual
 Speaker
 Suncica Canic, University of Houston
 Session Chair
 Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
We address certain challenges related to the analysis of moving boundary problems of parabolichyperbolic type. In particular, we focus on a fluidstructure interaction problem between the flow of an incompressible, viscous fluid and a multilayered poroelastic structure, which behaves as a compressible material. The coupled problem is described by the time dependent Stokes equations, which are coupled to the Biot equations over a poroelastic plate serving as an interface between the free fluid flow and the poroelastic structure. This problem was motivated by the design of a first bioartificial pancreas without the need for immunosuppressant therapy. We will show a recent constructive existence proof for a weak solution to this problem, and a (weakstrong) uniqueness result. This is one of only a handful of wellposedness results in the area of fluidporoelastic structure interaction problems. The mathematical reasons for this will be discussed, and the impact on the design of a bioartificial pancreas will be shown.

2:00  2:15 pm EDTBreakCoffee Break  Virtual

2:15  2:45 pm EDTStability of discontinuous solutions for inviscid compressible flowsVirtual
 Speaker
 Alexis Vasseur, University of Texas at Austin
 Session Chair
 Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
We will discuss some recent developments of the theory of a contraction with shifts to study the stability of discontinuous solutions of systems of equations modeling inviscid compressible flows. In a first result, in collaboration with Geng Chen and Sam Krupa, we provide some extensions of the Bressan theory for uniqueness of BV solutions in 1D. We show that for 2 × 2 systems, the technical condition, known as bounded variations on space like curve, is not needed for the uniqueness result. Moreover, we extend the result to a weak/BV stability result (in the spirit of the weak/strong principle of Dafermos) allowing wild perturbations fulfilling only the socalled strong trace property. In a second work in collaboration with MoonJin Kang, we consider the stability of 1D viscous shocks for the compressible NavierStokes equation, uniformly with respect to the viscosity (JEMS 21'). Thanks to the uniformity with respect to the viscosity, the result can be extended to the Euler equation (the associated inviscid model).This provides a stability result which holds in the class of wild perturbations of inviscid limits of solutions to NavierStokes, without any regularity restriction, not even the strong trace property (Inventiones 21'). This shows that the class of inviscid limits of NavierStokes equations is better behaved that the class of weak solutions to the inviscid limit problem. Finally, we will present a first multiD result obtained with MoonJin Kang and Yi Wang. We show the stability of contact discontinuities without shear, in the class of inviscid limits of compressible FourierNavierStokes equation. Note that it is still unknown whether nonuniqueness results can be obtained via convex integration for this special kind of singularity.

3:00  3:30 pm EDTConservation Laws in Biology: Two New ExamplesVirtual
 Speaker
 Marshall Slemrod, University of Wisconsin
 Session Chair
 Sigal Gottlieb, University of Massachusetts Dartmouth
Abstract
This talk is based on a paper with Matan Mussel ( NIH) that will appear in QAM. The talk provides two new applications of conservation laws in biology. The first is the application of the van der Waals fluid formalism for action potentials. The second is the application of the conservation laws of differential geometry (Gauss–Codazzi equations) to produce nonsmooth surfaces representing Endoplasmic Reticulum sheets.
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