ICMS »

Hyperbolic conservation laws and related analysis with applications

Sep 19, 2011 - Sep 23, 2011

ICMS 15 South College Street

Organisers

Name	Institution
Chen, Gui-Qiang	University of Oxford
Holden, Helge	Norwegian University of Science and Technology
Karlsen, Kenneth	University of Oslo

Discussing fundamental mathematical problems in nonlinear hyperbolic conservation laws arising in fluid mechanics, elasticity and relativity, including questions of existence, uniqueness, regularity, formation of singularities, and symptotic behaviour of solutions, this workshop presents an outstanding opportunity for mathematicians in the field to meet and survey recent analytical and numerical advances and to identify phenomena and theories likely to be important for future developments in the field.

Photo of delegates

Workshop delegates - Thursday 22 September [download the high resolution version for printing here]

Arrangements

Participation
Participation is by invitation only. Places for students and early career researches have been allocated.

UK Visas
If you are travelling from overseas you may require an entry visa. A European visa does not guarantee entry to the UK. Please use this link to the UK Visas site to find out if you need a visa and, if so, how to apply for one. If you do require a visa, ICMS can (on request) provide a signed invitation letter.

Venue
The workshop will take place in the Newhaven lecture theatre at ICMS, 15 South College Street, Edinburgh, Scotland. You may also find this map useful for the workshop.

Lecture theatre facilities
The lecture theatre is fitted with data projection, a visualiser (the new generation of over-head), smart-board and two blackboards. Please note that the blackboards and data projection cannot be used simultaneously, however, the projector and one board can be used together.

Wireless Access
ICMS at 15 South College Street, has wireless access throughout. On arrival at Registration you will be given instructions and a code for accessing the wireless network. For those without laptops, there are a number of computers available for you to check your emails. Delegates are welcome to make use of the public areas at 15 South College Street throughout the week. We suggest that all UK based participants sign up to eduroam at their home university.

Travel
Information about travel to the UK and Edinburgh is available here. Lothian buses charge 1.30 GBP for a single journey and 3.20 GBP for a day ticket. Please note that the exact fare is required and no change is given.

A taxi directly from the airport will cost approximately 18.00 to 20.00 GBP to the city centre for a one-way journey.

From the airport, you can take the airport bus Airlink to the city centre. It will cost you 3.50 GBP for a single ticket, or 6.00 GBP return. More details on the airport bus Airlink is available here. The bus will take you to Waverley Bridge (next to Waverley Railway Station).

If travelling by train, please note that Edinburgh has two railway stations - Waverley Railway Station being the main station and closest to the workshop venue ICMS. If you alight at Edinburgh Waverley, the workshop venue is an easy 10 minute walk over North and South Bridge map. The second railway station is called Haymarket and is at the West End of the city centre.

Please note that it is your responsibility to have adequate travel insurance to cover medical and other emergencies that may occur on your trip.

For Invited Participants Only: Accommodation
If your invitation included the offer of accommodation, then ICMS will arrange this and details will be in your letter of invitation. ICMS has reserved rooms for all invited delegates in the University of Edinburgh en suite accommodation. Accommodation is no more than a twenty minute walk from ICMS. Please indicate on your on-line registration form if you wish to take up the offer of these rooms or make your own arrangements.

Catering
Morning and afternoon refreshments will be provided on each day of the workshop.

Buffet lunch will be provided in the Chapterhouse on the ground floor of ICMS on Monday 19 September only.

The will be a workshop dinner on 22 September (Thursday).

Following the last talk on Monday 19 September and after the Knowledge Transfer event on Thursday 22 September, an informal wine reception, to which all all participants are invited, will take place in the Chapterhouse on the ground floor of ICMS.

Registration
Registration will take place in the morning on Monday 19 September at 15 South College Street, Edinburgh, EH8 9AA.

Conference Co-ordinator
Helene Frossling is the conference co-ordinator for this workshop.

Programme

Preliminary programme published 8 September 2011. Updated Monday 19 September

Monday 19 September

09.00 - 09.40	Registration and coffee
09.40 - 09.45	Welcome and introduction
09.45 - 10.30	Alberto Bressan (Penn State University) (talk given by Gui-Qiang Chen) Survey/Introductory lecture I: 1-D hyperbolic conservation laws
10.30 - 11.00	Coffee and tea (served in the Chapterhouse)
11.00 - 11.45	Alberto Bressan (Penn State University) (talk given by Carlotta Donadello) Survey/Introductory lecture II: 1-D hyperbolic conservation laws
11.50 - 12.35	Laura Caravenna (University of Oxford) SBV regularity of Entropy Solutions to genuinely non-linear, strictly hyperbolic systems of conservation laws
12.35 - 14.30	Lunch served in the Chapterhouse
14.30 - 15.15	Alberto Bressan (Penn State University) (talk given by Carlotta Donadello) Survey/Introductory lecture III: 1-D hyperbolic conservation laws
15.15 - 15.45	Coffee and tea
15.45 - 16.30	Konstantina Trivisa (University of Maryland) Hyperbolic balance laws: rate of convergence for vanishing viscosity approximations
16.35 - 17.20	Trygve Karper (University of Maryland) Preventing blow-up in the Keller-Segel model of chemotaxis

Tuesday 20 September

09.00 - 09.45	Denis Serre (Ecole Normale Supérieure de Lyon) Survey/Introductory lecture I: Multi-dimensional systems of conservation laws
09.50 - 10.35	Mikhail Feldman (University of Wisconsin-Madison) Shock reflection, free boundary problems, and degenerate elliptic equations
10.35 - 11.05	Coffee and tea served in the Chapterhouse
11.05 - 11.50	Denis Serre (Ecole Normale Supérieure de Lyon) Survey/Introductory lecture II: Multi-dimensional systems of conservation laws
11.55 - 12.40	Hairong Yuan (East China Normal University) Uniqueness of steady transonic shocks in two-dimensional compressible Euler flows
12.40 - 14.30	Lunch break
14.30 - 15.15	Steve Skholler (University of California, Davis) Free boundary problems for the compressible Euler equations
15.15 - 15.45	Coffee and tea served in the Chapterhouse
15.45 - 16.30	Wladimir Neves (Federal University of Rio de Janeiro) The multidimensional Muskat problem

Wednesday 21 September

09.00 - 09.45	Mihalis Dafermos (University of Cambridge) Survey/Introductory lecture I: The Cauchy problem in General Relativity
09.50 - 10.35	Marshall Slemrod (University of Wisconsin) Higher dimensional isometric embedding
10.35 - 11.05	Coffee and tea served in the Chapterhouse
11.05 - 11.50	Mihalis Dafermos (University of Cambridge) Survey/Introductory lecture II: The Cauchy problem in General Relativity
11.55 - 12.40	Alexis Vasseur (University of Texas at Austin) Relative entropy and the stability of shocks for systems of conservation law
	Free afternoon

Thursday 22 September

09.00 - 09.45	Sylvie Benzoni-Gavage (Université Claude Bernard Lyon 1) Weakly nonlinear surface waves
09.50 - 10.35	Tai-Ping Liu (Stanford University and Academia Sinica, Taipei) Boundary relation for dissipative systems
10.35 - 11.05	Coffee and tea served in the Chapterhouse
11.05 - 11.50	Hermano Frid (IMPA) Spatially periodic solutions for a non-isothermal polytropic gas flow
11.55 - 12.40	Boris Andreianov (Université de Franche-Comté) Well-posedness and approximation of a "particle-in-Burgers" model
12.40 - 14.30	Lunch break
14.30 - 15.15	Dehua Wang (University of Pittsburgh) Global solutions for liquid crystals
15.15 - 15.45	Coffee and tea served in the Chapterhouse
15.45 - 16.30	Raimund Bürger (Universidad de Concepción) On nonlocal conservation laws modelling sedimentation
16.35 - 17.20	Wei Xiang (University of Oxford) Shock diffraction problem for the two dimensional nonlinear wave system and potential flow equation
	Public lecture
18.00 - 19.00	Sir John Ball (University of Oxford) What can mathematics say about liquid crystals?
19.00 - 19.45	Wine reception
20.00	Workshop dinner at The Magnum Restaurant (for workshop participants)

Friday 23 September

09.50 - 10.35	Pierangelo Marcati (University of L' Aquila) Analysis of defect measures and correctors for plasma oscillations
10.35 - 11.10	Coffee and tea served in the Chapterhouse
11.10 - 11.55	Constantine Dafermos (Brown University) Stability in elastodynamics
12.00 -	Lunch
13.30 - 15.00	Free discussion alt. more talks

Presentations:

Presentation Details
Andreianov, Boris
Well-posedness and approximation of a "particle-in-Burgers" model
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We discuss the one-dimensional fluid-particle interaction model, formulated by F. Lagoutiere, N. Seguin and T. Takahashi in 2008. It consists of the Burgers equation with singular source term localized along the particle path, and of an ODE governing the particle location. The singular Burgers equation can be interpreted within the setting of conservation laws with discontinuous flux: indeed, it can be seen as two conservation laws coupled (in a non-conservative way) across the interface created by the particle path. We prove existence and uniqueness of the appropriately defined entropy solutions of the problem, and describe and analyze numerical schemes for their approximation.
Ball, Sir John
What can mathematics say about liquid crystals? (Public lecture.)
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Liquid crystals represent a vast and diverse class of materials which are intermediate between isotropic liquids and crystalline solids. Liquid crystal ordering is found in a wide variety of systems, ranging from fluids made up of simple rods, polymers, elastomers and biological organisms. Liquid crystals have a multitude of applications, notably those in flat panel display technology, which has fundamentally impacted modern life. The lecture will describe these fascinating materials and what mathematics, in particular the study of partial differential equations, can tell us about them.
Benzoni-Gavage, Sylvie
Weakly nonlinear surface waves
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In scale invariant boundary value problems, linear surface waves of finite energy like the Rayleigh waves in elasticity are associated with modulated waves in the weakly nonlinear regime, which are governed by an amplitude equation that is a nonlocal generalization of Burgers' equation. In this talk, we will make the connection between the structure and stability properties of the amplitude equation and those of the original, fully nonlinear problem. We will consider two types of problems: first-order hyperbolic systems of PDEs and higher-order Hamiltonian PDEs, the prototypes of which are the Euler equations of gaz dynamics and the equations of elastodynamics. This is a joint work with Jean-François Coulombel.
Bürger, Raimund
On nonlocal conservation laws modelling sedimentation
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A nonlinear conservation law with nonlocal flux arises from a one-dimensional kinematic sedimentation model if we assume that the solids settling velocity or the slip velocity depends on the local solids fraction in a finite neightborhood via convolution with a kernel. Uniqueness of the resulting nonlocal equation is established via an entropy solution concept. Existence is shown by convergence of a difference-quadrature scheme. Numerical examples are presented and the connection with dispersive local PDEs is discussed.
Caravenna, Laura
SBV regularity of Entropy Solutions to genuinely non-linear, strictly hyperbolic systems of conservation laws
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This talk concerns a regularity result for entropy solutions of strictly hyperbolic systems of conservation laws. By Glimm's estimates, for small BV initial data the entropy solution belongs to the space of functions of bounded variation for all times. However, the space BV does not rule out a nasty fractal behavior of the derivative, as for the Cantor-Vitali function. Special functions of bounded variation are instead those whose gradient is a measure consisting only of a jump part plus an absolutely continuous part. Ambrosio-De Lellis have shown in the case of a single equation with uniformly convex flux that the entropy solution is indeed a special function of bounded variation. We will illustrate in the talk this situation. We will then show that the same behavior takes place also in the case of systems with only genuinely non-linear characteristic fields (joint work with S. Bianchini).
Dafermos, Constantine
Stability in elastodynamics
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The lecture will discuss how "convexity" of the strain energy function affects the stability of motion in elastodynamics.
Feldman, Mikhail
Shock reflection, free boundary problems, and degenerate elliptic equations.
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We discuss existence and regularity of global solutions to shock reflection for potential flow. Self-similar potential flow equation is a nonlinear equation of mixed elliptic-hyperbolic type. Main issues in analysis of shock reflection include estimates of solution near sonic line where the type of equation changes, and control of geometry of free boundary. Open problems will also be discussed. The talk is based on joint works with Gui-Qiang Chen.
Karper, Trygve
Preventing blow-up in the Keller-Segel model of chemotaxis
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We consider the simplified inviscid Keller-Segel model of chemotaxis. Our starting point is that solutions concentrate at single points in finite time (dirac masses). In some cases, this blow-up might be undesirable. For instance, there might be a limit to the number of bacteria that can occupy the same physical space. Numerous modifications have been proposed to avoid or delay blow-up. The most typical and studied modification consists of adding diffusion (local repulsion) to counteract the aggregation. In this talk, we will examine several of the proposed modifications. Moreover, we will present a new modification that incorporates a maximal density constraint and prove that the resulting model is well-defined.
Liu, Tai-Ping
Boundary relation for dissipative systems
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With Shih-Hsien Yu we study the relation of boundary values and their differentials for dissipative systems. We consider various physical systems to demonstrate the techniques of inverting Lapalce-Fourier transforms used in our approach.
Marcati, Pierangelo
Analysis of defect measures and correctors for plasma oscillations
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(joint with D. Donatelli) We perform a rigorous analysis of the quasineutral limit for a hydro-dynamical model of a viscous plasma represented by the Navier Stokes Poisson system in 3 -D. We show that as Debye length tends to 0 the velocity field strongly converges towards an incompressible velocity vector field and the density fluctuation weakly converges to zero. In general the limit velocity field cannot be expected to satisfy the incompressible Navier Stokes equation due to the presence of high frequency oscillations, which strongly affects the quadratic nonlinearities and because of the self interacting wave packets. We shall provide a detailed mathematical description of the convergence process by using microlocal defect measures and by an explicit correctors analysis. Moreover we will be able to identify an explicit pseudo parabolic pde satisfied by the leading correctors terms. Our results include all the previous results in literature, in particular we show that the formal limit holds rigorously in the case of well prepared data.
Neves, Wladimir
The multidimensional Muskat problem
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In this talk, we study the multidimensional Muskat problem. We show the solvability of the initial-boundary value problem to a proposed generalized Buckley-Leverett system. Moreover, we discuss some important questions concerning singular limits of the proposed model.
Serre, Denis
Multi-dimensional systems of conservation laws


Slemrod, Marshall
Higher dimensional isometric embedding
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The problem of isometric embedding an n dimensional Riemannian manifold into n(n+1)/2 Euclidean space is a classic issue in differential geometry. In this talk I will outline recent results on this topic that I have done with my co-researchers J. Clelland, G-Q Chen, D. Wang, D. Yang. The key idea is to relate the problem to continuum mechanics analogies and then obtain crucial " entropy" equalities for the relevant balance laws.
Trivisa, Konstantina
Hyperbolic balance laws: rate of convergence for vanishing viscosity approximations
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Hyperbolic balance laws result from the balance laws of continuum physics and has a variety of applications in elasticity, gas dynamics and materials science. In this work, we consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers’s equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals. Furthermore, results on the convergence rate for vanishing viscosity approximations for hyperbolic balance laws are presented. This is joint work with C. Christroforou.
Vasseur, Alexis
Relative entropy and the stability of shocks for systems of conservation laws
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We develop a theory based on relative entropy to show the uniqueness and L^2 stability (up to a translation) of extremal entropic Rankine-Hugoniot discontinuities for systems of conservation laws (typically 1-shocks, n-shocks, 1-contact discontinuities and n-contact discontinuities of large amplitude) among bounded entropic weak solutions having an additional 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.
Wang, Dehua
Global solutions for liquid crystals
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Global solutions with various regularities will be discussed for both incompressible and compressible flows of liquid crystals.
Xiang, Wei
Shock diffraction problem for the two dimensional nonlinear wave system and potential flow equation
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The vertical shock which initially separates two piecewise constant Riemann datas, passes the wedge from left to right, then shock diffraction phenomena occur and the incident shock becomes a transonic shock. Here we study this problem on the nonlinear wave system. The existence and the optimal regularity across the sonic circle of the solutions to this problem are discussed. Finally we give some remarks on this problem for the potential flow equation.
Yuan, Hairong
Uniqueness of steady transonic shocks in two-dimensional compressible Euler flows
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This is a joint work with Beixiang Fang (Shanghai Jiaotong University) and Li Liu (Shanghai Institute of Foreign Trade). We prove that for the two-dimensional steady complete compressible Euler system, with given uniform upcoming supersonic flows, the following three fundamental flow patterns (special solutions) in gas dynamics involving transonic shocks are all unique in the class of piecewise $C1$ smooth functions, under appropriate conditions on the downstream subsonic flows: (1) the normal transonic shocks in a straight duct with finite or infinite length, after fixing a point the shock-front passing through; (2) the oblique transonic shocks attached to an infinite wedge; (3) a flat Mach configuration containing one supersonic shock, two transonic shocks, and a contact discontinuity, after fixing the point the four discontinuities intersect. These special solutions are constructed traditionally under the assumption that they are piecewise constant, and they have played important roles in he studies of mathematical gas dynamics. Our results show that the assumption of piecewise constant can be replaced by some more weaker assumptions on the downstream subsonic flows, which are sufficient to uniquely determine these special solutions. Mathematically, these are uniqueness results on solutions of free boundary problems of a quasi-linear system of elliptic-hyperbolic composite-mixed type in bounded or unbounded planar domains, without any assumptions on smallness. The proof relies on an elliptic system of pressure $p$ and the tangent of the flow angle $w=v/u$ obtained by decomposition of the Euler system in Lagrangian coordinates, and a newly developed method for the $L^{infty}$ estimate that is independent of the free boundaries, by combining the maximum principles of elliptic equations, and careful analysis of shock polar applied on the (maybe curved) shock-fronts.

Participants

Name	Institution
Andreianov, Boris	Université de Franche-Comté
Ball, Sir John	University of Oxford
Benzoni-Gavage, Sylvie	Université Claude Bernard Lyon 1
Bürger, Raimund	Universidad de Concepción
Caravenna, Laura	University of Oxford
Chen, Gui-Qiang	University of Oxford
Coutand, Daniel	Heriot-Watt University
Dafermos, Constantine	Brown University
Dafermos, Mihalis	Univeristy of Cambridge
Ding, Min	University of Oxford
Donadello, Carlotta	Université de Franche-Comté
Feldman, Mikhail	University of Wisconsin-Madison
Frid, Hemano	IMPA
Grunert, Katrin	Norwegian University of Science and Technology
Han, Weiwei	Donghua University
Holden, Helge	Norwegian University of Science and Technology
Karlsen, Kenneth	University of Oslo
Karper, Trygve	University of Maryland
Knops, R J	Heriot-Watt University
Kukreja, Vaibhav	Northwestern University
Kumar K, Sudarshan	Tata Institute of Fundamental Research
Kursungecmez, Hatice	University of Edinburgh
Liu, Tai-Ping	Stanford University and Academia Sinica, Taipei
Marcati, Pierangelo	University of L' Aquila
Neves, Wladimir	Federal University of Rio de Janeiro
Perepelitsa, Mikhail	University of Houston
Raynaud, Xavier	Centre of Mathematics for Applications
Serre, Denis	Ecole Normale Supérieure de Lyon
Shkoller, Steve	University of California, Davis
Slemrod, Marshall	University of Wisconsin
Stevens, Ben	University of Oxford
Trivisa, Konstantina	University of Maryland
Vasseur, Alexis	University of Texas at Austin
Wang, Dehua	University of Pittsburgh
Xiang, Wei	University of Oxford
Yuan, Hairong	East China Normal University
Zhang, Lei	University of Oxford