We provide an extension of the theory of flows in Euclidean spaces associated to wekly differentiable velocity fields, replacing the Euclidean state space with the space of probability measure. This way, the continuity equation can be viewed as an ODE in the space of probability measures and we provide some well-posedness and stability results. We illustrate, as an example and a basic motivation for the development of the theory, the problem of convergence of Wigner transforms in Quantum Mechanics.

This set of lectures will provide a basic introduction to hyperbolic systems of conservation laws in one space dimension. The main topics covered will be:

1. Meaning of the conservation equations and definition of weak solutions.



2. Shocks, Rankine-Hugoniot equations and admissibility conditions.



3. The Riemann problem. Wave interaction estimates.



4. Weak solutions to the Cauchy problem with small BV data.



5. Uniqueness and continuous dependence.



6. Vanishing viscosity approximations.



7. Some counter-examples to global existence, uniqueness, and continuous dependence of solutions, when some key hypotheses are removed.

This series of four lectures will provide an introduction to multidimensional conservation laws, along with a perspective on a list of open problems. The main topics will include:

1. Prototypes and Basic Features/Phenomena



2. Multidimensional Models



3. Multidimensional Steady Problems



4. Multidimensional Self-Similar Problems



5. Compressible Vortex Sheets and Related Problems



6. Divergence-Measure Fields and Hyperbolic Conservation Laws

Lecture References:

1. Gui-Qiang Chen, Euler Equations and Related Hyperbolic Conservation Laws, In: Handbook of Differential Equations: Evolutionary Differential Equations, Vol. 2, pp. 1-104, 2005, Eds. C.~M. Dafermos and E. Feireisl, Elsevier: Amsterdam, Netherlands



2. Gui-Qiang Chen and Mikhail Feldman, Shock Reflection-Diffraction and Multidimensional Conservation Laws, In: Proceedings of the 2008 Hyperbolic Conference: Theory, Numerics, and Application, AMS: Providence, 2009.



3. Gui-Qiang Chen, Monica Torres, and William Ziemer, Measure-Theoretical Analysis and Nonlinear Conservation Laws, Pure Appl. Math. Quarterly, 3 (2007), 841-879 (To Leon Simon on his 60th birthday).



4. Gui-Qiang Chen, Marshall Slemrod, and Dehua Wang, Conservation Laws: Transonic Flow and Differential Geometry, In: Proceedings of the 2008 Hyperbolic Conference: Theory, Numerics, and Application, AMS: Providence, 2009.

*You may directly download them from http://www.math.northwestern.edu/~gqchen/preprints/

Joint work with with Chunjing Xie. We prove global existence and asymptotic behavior of classical solutions for two dimensional inviscid Rotating Shallow Water system with small initial data subject to the zero-relative-vorticity constraint. Such global existence is in contrast with the finite time breakdown of compressible Euler equations in two and three dimensions. A key step in our proof is reformulation of the problem into a symmetric quasilinear Klein-Gordon system, which is then studied with combination of the vector field approach and the normal forms. We also probe the case of general initial data and reveal a lower bound for the lifespan that is almost inversely proportional to the size of the initial relative vorticity.

Given a model based on a conservation law, we study how the solution depends from the initial/boundary datum, from the flow and from various constraints. With this tool, several control problems can be addressed and the existence of an optimal control can be proved. Models describing escape dynamics of pedestrians, traffic at toll gates, open canals management and fluid flow in gas pipelines fall within this framework. In particular, a necessary condition for optimality is obtained, which applies to a supply chain model.

I am planning to convey a feel for the area by touching upon its history, its general features, the current directions, etc., paving the way for the more technical lectures by others.

We consider two gasdynamic contexts: an isentropic context and, respectively, an anisentropic context of a special type. Some [multidimensional] classes of regular solutions [simple waves solutions and regular interactions of simple waves solutions] can be constructively associated to the isentropic context, in presence of an ad hoc requirement of genuine nonlinearity, via a Burnat type ["algebraic"] approach [centered on a duality connection between the hodograph character and the physical character]. The "algebraic" construction fails generally [for geometrical and/or physical reasons] in the mentioned anisentropic context. Incidentally, in this case the "algebraic" construction can be replaced by a Martin type ["differential"] approach [centered on a Monge-Ampere type representation]. Some classes [unsteady 1D; steady supersonic] of regular solutions [pseudo simple waves solutions, regular interactions of pseudo simple waves solutions] can be constructed in this case via a "linearized" version of the "differential" approach. Some contrasts and consonances are considered, in a classifying parallel, between the two approaches ["algebraic", "differential"] and the two classes [isentropic, anisentropic] of regular solutions respectively associated.

This is a joint work with Marina Ileana Dinu.

We consider a piecewise smooth solution to a scalar conservation law, with possibly interacting shocks. After the interactions have taken place, vanishing viscosity approximations can still be represented by a regular expansion on smooth regions and by a singular perturbation expansion near the shocks, in terms of powers of the viscosity coefficient.

In one space dimension, the Godunov scheme can only converge to entropy solutions, for which uniqueness theorems in various classes are known. We present an example in 2d where the Godunov scheme converges, depending on the grid, either to an exact entropy solution or to a second, numerical solution, which is also produced by all other numerical schemes tested. We propose that entropy solutions are not unique in 2d. The observation also yields an explanation for the "carbuncle phenomenon", an instability in numerical calculations of shock waves. Thus fundamental consequences both for numerics and theory.

Singular limits in the theory of partial differential equations represent a class of problems, where some parameters in the equations become small or infinitely large. We discuss the singular limits arising in the scale analysis of energetically isolated fluid systems. In particular, the following topics are addressed:

• functional analytic framework and the basic ideas of the mathematical theory of continuum fluid mechanics;



• thermodynamic stability and equilibrium states;



• low Mach number limits and propagation of acoustic waves in thermally conducting viscous fluids

In this talk we will start with discussion of shock reflection phenomena. Then we describe recent results on existence and stability of global solutions to regular shock reflection for potential flow for all wedge angles up to the sonic angle, and discuss the techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the sonic arc). We will discuss techniques to handle such free boundary problems and degenerate elliptic equations. This is a joint work with Gui-Qiang Chen.

In this talk we consider a conservation law (or a system of conservation laws) on a network consisting in a finite number of arcs and vertices. This setting is justified by various applications, such as car traffic, gas pipelines, data networks, supply chains, blood circulation and so on. The key point in the extension of conservation laws on networks is to define solutions at vertices. Indeed, it is sufficient to define solutions only for Riemann problems at vertices, i.e. Cauchy problems with constant initial data in each arc of the junction. We present some different possibilities to produce solutions to Riemann problems at vertices. Moreover we consider the general Cauchy problem on the network. We explain how to prove existence of a solution both in the scalar case and in the case of systems. In particular, for the scalar case, we introduce general properties on Riemann solvers at vertices, which permit to have existence of solutions for the Cauchy problem.

Numerical approximation of fluid equations are reviewed. We identify numerical mass diffusion as a characteristic problem in most simulation codes. This fact is illustrated by an analysis of fluid mixing flows. In these flows, numerical mass diffusion has the effect of over regularizing the solution.

A number of startling conclusions have recently been observed. For a flow accelerated by multiple shock waves, we observe an interface between the two fluids proportional to Delta x-1, that is occupying a constant fraction of the available mesh degrees of freedom. This result suggests (a) nonconvergence for the unregularized mathematical problem or (b) nonuniqueness of the limit if it exists, or (c) limiting solutions only in the very weak form of a space time dependent probability distribution.

The cure for this pathology is a regularized solution, in other words inclusion of all physical regularizing effects, such as viscosity and physical mass diffusion.

In other words, the amount of regularization of an unstable flow is of central importance. Too much regularization, with a numerical origin, is bad, and too little, with respect to the physics, is also bad.

At the level of numerical modeling, the implication from this insight is to compute solutions of the Navier-Stokes, not the Euler equations. Resolution requirements for realistic problems make this solution impractical in most cases. Thus subgrid transport processes must be modeled, and for this we use dynamic models of the turbulence modeling community. In the process we combine and extend ideas of the capturing community (sharp interfaces or numerically steep gradients) with conventional turbulence models, usually applied to problems relatively smooth at a grid level.

The numerical strategy is verified with a careful study of a 2D Richtmyer-Meshkov unstable turbulent mixing problem. We obtain converged solutions for such molecular level mixing quantities as a chemical reaction rate. The strategy is validated (comparison to laboratory experiments) through the study of three dimensional Rayleigh-Taylor unstable flows.

In applications of flow in gas and water networks the 1d p-system is used to describe the flow in pipes. The dynamics of each pipe is coupled at a node through coupling conditions inducing boundary conditions. We study the well-posedness of these initial boundary value problems of 2×2 balance laws for a set of general possibly time-dependent coupling conditions.

We show that, for a model system of compressible fluid flow in the upper half space of the plane, curves which intersect the boundary and across which the initial density is discontinuous become tangent to the boundary instantaneously in time. This result is closely related to the instantaneous formation of cusps in two-dimensional incompressible vortex patches.

We consider the KdV equation used for modeling, e.g., water waves in a narrow channel. The time evolution is described by a quadratic (Burgers') plus a linear (Airy) term. Since the evolution of each of these terms is quite different, it is natural to ask whether the concatenation of the evolution operators for each term yields an approximation to the evolution operator for the sum. This strategy has been used with good results when designing numerical methods for the KdV equation, but without rigourous convergence proofs. The aim of this talk is to present a first step in this direction, and show convergence of operator splitting for sufficiently regular initial data. (Joint work with N. H. Risebro, K. H. Karlsen, T. Tao.)

The global strong solutions to the multi-dimensional viscoelastic fluids and liquid crystals are discussed. Here, a strong solution is a solution in W^2,q satisfying the equations almost everywhere.

The Kriess-Sakamoto theory for the well-posedness of hyperbolic IBVPs and the Majda theory for shock-wave stability apply under the assumption that a suitable Lopatinski condition holds uniformly. The failure of uniformity is associated with the presence of surface waves on the boundary or discontinuity. We will derive asymptotic equations for "genuine" nonlinear surface waves that decay exponentially away from the surface, such as Rayleigh waves in elasticity or surface waves on a tangential discontinuity in MHD. We will give a short-time existence theorem for smooth solutions of the asymptotic equations under a "tameness" condition on the interaction coefficients between the Fourier components of the surface wave, which prevents the loss of derivatives. We will also show numerical solutions of the asymptotic equations that illustrate singularity formation on the boundary (a mechanism which differs from shock formation in the interior).

We study a Riemann problem for the unsteady transonic small disturbance equation that results in a diverging rarefaction problem. We write the problem in self-similar coordinates and obtain a free boundary value problem with equations that change type (hyperbolic-elliptic). We summarize the main ideas and present the main features of the problem. The flow in the hyperbolic part can be described as a solution of a degenerate Goursat boundary problem, the interaction of the rarefaction wave with the subsonic region is illustrated and the subsonic flow is shown to satisfy a second order degenerate elliptic boundary problem with mixed boundary conditions. This is joint work with Jun Chen and Cleopatra Christoforou (University of Houston).

Motivated by the problem of symmetric collapsing gas-dynamical shocks we present a scalar toy model that captures blowup of focusing waves. This model is simple enough to allow for explicit calculations, and we study some solutions in detail. As a scalar model it does not describe reflection of waves and this necessitates a new concept of weak solutions.

Returning to gas-dynamics we review a part of the extensive literature on compressible flow with symmetry, and collapsing shocks in particular. We outline an approach to study possible blowup for collapsing shocks.

Conservation laws, together with the Gauss law for electrostatics, have been used to model charge transport in solid state semiconductors and in electrolytes for several decades. The determination of the current density is an important aspect of the modeling. In applications to ion channels, and to electrodiffusion more generally, there has been recent interest in the effects of the ambient fluid on current density. We discuss the mathematical model for this case: the Poisson-Nernst-Planck/Navier-Stokes model. The Cauchy problem was investigated by the speaker in [Transport Theory Statist. Phys. 31 (2002), 333--366], where a local existence-uniqueness theory was demonstrated, based upon Kato's framework for evolution equations. In this talk, the proof of existence of a global distribution solution for the model is discussed, in the case of the initial-boundary value problem. Connection of the above analysis to significant applications is also discussed.

Using periodic Evans function expansions, we derive geometric criterion for the modulational instability, i.e. spectral instability to long-wavelength perturbations, of periodic traveling waves of the generalized KdV equation. Such techniques have also recently proven useful for analyzing spectral stability to long-wavelength transverse perturbations, as well as studying nonlinear stability to periodic perturbations. Using standard elliptic function techniques, we can explicitly calculate the necessary geometric information for polynomial nonlinearities: we will only present the results for KdV and modified KdV equations here.

This is joint work with Jared. C. Bronski (University of Illinois at Urbana-Champaign).

The study of self-similar solutions of multidimensional conservation laws leads to systems of equations that change type. Change of type occurs either across a transonic shock or at a sonic line. Often the sonic line appears as a free boundary in the formulation of the problem. Some recent numerical (and experimental) discoveries of a new kind of shock reflection ('Guderley Mach reflection') lead to interesting and still unresolved questions concerning the nature of the self-similar solutions in this generic case.

In this talk, I will present some analysis of a simple model for this phenomenon, using the transonic small disturbance equation. The simplified problem seems amenable to analysis, but we are just beginning to make progress. This is a report on current joint work with Allen Tesdall and Kevin Payne.

We present a numerical method for the compressible Euler equations in 3 space dimensions where we combine adaptive mesh refinement with subgrid scale modelling. This increases efficiency and accuracy when computing turbulent flows. We apply this to astrophysical problems.

I will discuss the existence and properties of small-scale dependent shock waves to nonlinear hyperbolic systems, with an emphasis on the theory of nonclassical entropy solutions involving undercompressive shocks. Regularization-sensitive structures often arise in continuum physics, especially in flows of complex fluids or solids. The so-called kinetic relation was introduced for van der Waals fluids and austenite-martensite boundaries (Abeyaratne, Knowles, Truskinovsky) and nonlinear hyperbolic systems (LeFloch) to characterize the correct dynamics of subsonic phase boundaries and undercompressive shocks, respectively. The role of a single entropy inequality is essential for these problems and is tied to the regularization associated with higher-order underlying models –which take into account additional physics and provide a description of small-scale effects. In the last fifteen years, analytical and numerical techniques were developed, beginning with the construction of nonclassical Riemann solvers, which were applied to tackle the initial-value problem via the Glimm scheme. Total variation functionals adapted to nonclassical entropy solutions were constructed. On the other hand, the role of traveling waves in selecting the proper shock dynamics was stressed: traveling wave solutions (to the Navier-Stokes-Korteweg system, for instance) determine the relevant kinetic relation –as well as the relevant family of paths in the context of nonconservative systems. Several physical applications were pursued: (hyperbolic-elliptic) equations of van der Waals fluids, model of thin liquid films, generalized Camassa-Holm equations, etc. Importantly, finite difference schemes with controled dissipation based on the equivalent equation were designed and the corresponding kinetic functions computed numerically. Consequently, `several shock wave theories' are now available to encompass the variety of phenomena observed in complex flows.

References:

1993: P.G. LeFloch, Propagating phase boundaries. Formulation of the problem and existence via the Glimm scheme, Arch. Rational Mech. Anal. 123, 153–197.

1997: B.T. Hayes and P.G. LeFloch, Nonclassical shocks and kinetic relations. Scalar conservation laws, Arch. Rational Mech. Anal. 139, 1–56.

2002: P.G. LeFloch, Hyperbolic Systems of Conservation Laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zurich, Birkhauser.

2004: N. Bedjaoui and P.G. LeFloch, Diffusive-dispersive traveling waves and kinetic relations. V. Singular diffusion and dispersion terms, Proc. Royal Soc. Edinburgh 134A, 815–844.

2008: P.G. LeFloch and M. Mohamadian, Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations, J. Comput. Phys. 227, 4162–4189.

We consider scalar nonviscous conservation laws with strictly convex flux in one space dimension, and we investigate the behavior of bounded L² perturbations of shock wave solutions to the Riemann problem using the relative entropy method. We show that up to a time-dependent translation of the shock, the L² norm of a perturbed solution relative to the shock wave is bounded above by the L² norm of the initial perturbation.

Elastic materials exhibit qualitatively different responses to different kinematic boundary conditions or body forces. As a first step towards understanding the related evolutionary problem, one studies the minimizers of an appropriate nonlinear elastic energy functional.

We shall give an overview of recent results, rigorously deriving 2d elasticity theories for thin 3d shells around mid-surfaces of arbitrary geometry. One major ingredient is the study of Sobolev spaces of infinitesimal isometries on surfaces, their density and matching properties. Another one relates to the non-Euclidean version of 3d nonlinear elasticity, conjectured to explain the mechanism for spontaneous formation of non-zero stress equilibria in growing tissues (leaves, flowers). Here, we prove a Gamma-convergence result, and as a corollary, we obtain new conditions for existence of isometric immersions of 2d Riemannian metrics into 3d space.

The study of shock wave theory for hyperbolic and viscous conservation laws is a good starting point for the study of certain aspects of kinetic theory. It offers different perspective and techniques than those coming from statistical mechanics. The main difference is the emphasis on the fluid dynamical phenomena. However, it is also important to realize that kinetic theory offers more than gas dynamics, particularly when it comes to the shock, initial and boundary layers. Also, depending on the physical situations, different fluid dynamics equations are derived from the Boltzmann equation. We will comment on these issues and present some recent results on invariant manifolds for stationary Boltzmann equation.

I will present some results on the existence and nonlinear stability for the rotating star solutions which are axi-symmetric steady-state solutions of the compressible isentropic Euler-Poisson equations in 3 spatial dimensions. We then apply these results to rotating white dwarf stars to show its dynamical stability when the total mass is less than a critical mass, which is related to the "Chandrasekhar"limit in astrophysics. This is a joint work with Joel Smoller.

It will be reviewed briefly various physical models leading to a description based on Quantum Hydrodynamics: Superfluidity, BEC, superconductivity, semiconductors and there will be recalled various derivations of the PDE system. The main result (joint with P. Antonelli) shows the global existence of "irrotational" weak solutions with the sole assumption of finite energy, without any smallness or any further smoothness of the initial data. The approach is based on various tools, namely the wave functions polar decomposition, the construction of approximate solution via a fractional steps method which iterates a Schrödinger Madelung picture with a suitable wave function updating mechanism. Therefore several a priori bounds of energy, dispersive and local smoothing type, allow to prove the compactness of the approximating sequences. A different approach can be used to study small disturbances of subsonic (in the QHD sense) steady states. Some improvements may are shown to be possible in the 2-D analysis.

We prove the global existence of regular solutions to the water waves problem in 3D. The proof is based on the combinaison of energy estimates and dispersive estimates.

This is a joint work with Pierre Germain and Jalal Shatah.

We will consider the problem of the motion of several rigid bodies in viscous non-Newtonian heat fluid in bounded domain in three dimensional situation.Using penalization method developed by Conca, San Martin, Tucsnak and Starovoitov we have shown the existence of weak solution and moreover we show that for certain non-newtonian fluids there are no collisions among bodie or body and boundary of domain.

We present our recent results on one- and multi-dimensional asymptotic stability of noncharacteristic boundary layers in gas dynamics (or a more general class of hyperbolic--parabolic systems) with suction- or blowing-type boundary conditions. The linearized and nonlinear stability is established for layers with arbitrary amplitudes, under the assumption of strong spectral, or uniform Evans, stability. The latter assumption has been verified for small-amplitude layers in one-d case by various authors using energy estimates and in multi-d case by Guès, Métivier, Williams, and Zumbrun. For large-amplitude layers, it may be efficiently checked numerically by a combination of asymptotic ODE estimates and numerical Evans function computations.

This is a joint work with Kevin Zumbrun.

We describe the approach through which various elastic shell models are rigorously derived from the 3D nonlinear elasticity theory. Some results and a conjecture on the limiting theories are presented. Spaces of weakly regular isometries or infinitesimal isometries of surfaces arise in this context. Important problems regarding these spaces include rigidity, regularity and density of smooth mappings.

This project is partly a collaboration with Marta Lewicka and Maria-Giovanna Mora.

We propose a new fourth-order non-oscillatory central scheme for computing approximate solutions of hyperbolic conservation laws. A piecewise cubic polynomial is used for the spatial reconstruction and for the numerical derivatives we choose genuinely fourth-order accurate non-oscillatory approximations. The solution is advanced in time using natural continuous extension of Runge-Kutta methods. Numerical tests on both scalar and gas dynamics problems confirm that the new scheme is non-oscillatory and yields sharp results when solving profiles with discontinuities. Experiments on nonlinear Burgers’ equation indicate that our scheme is superior to existing fourth-order central schemes in the sense that the total variation (TV) of the computed solutions are closer to the total variation of the exact solution.

Several models have been proposed in order to describe cell communities self-organisation. One of them consists in coupling a multidimensional scalar conservation law with an elliptic equation which gradient determines the flux in the conservation law.

In dimension larger than 1, the model looses all nice properties of hyperbolic conservation laws: no contraction property, no BV bound, no regularizing effect. That is the reason why our approach for existence of solutions is based on the kinetic formulation. We recall how weak limits can be handled with this tool and strong convergence follows from uniqueness.

In the case at hand, the specific nonlinearity creates an additional defect measure. Fine analysis of properties of this measure provides us with the lacking information to prove uniqueness and deduce that the weak limit still satisfies the system.

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain. We show existence of a strong solution that is a small perturbation of a constant flow (v^* = (1,0,0), ρ^* = 1). We also show that this solution is unique in the class of small perturbations of (v^*,ρ^*). The nonlinear term in the continuity makes it impossible to apply a fixed point argument. Therefore in order to show the existence of the solution we use a method of successive approximations.

Proceeding from the method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we construct a solution to the Riemann problem for the pressureless gas dynamics and sticky particles system. We analyze the difference in the behavior of discontinuous solution for these two models and relations between them.

In his celebrated thesis, S. Kawashima gave a framework for the analysis of the Cauchy problem for nonlinear viscous systems of conservation laws. Some assumptions are quite natural, while other ones are mysterious and require cumbersome calculations. We clarify the situation, by introducing a set of assumption which is natural and very easy to verify. We derive an existence and uniqueness result of strong solutions, in a slightly larger class than the one known before.

Joint work with Debora Amadori (Università degli Studi dell'Aquila).

We study a 2×2 system of balance laws that describes the evolution of a granular material (avalanche) flowing downhill. The original model was proposed by Hadeler and Kuttler.

We first consider an initial-boundary value problem, where at the boundary the flow of the incoming material is assigned. For this problem we prove the global existence of BV solutions for a suitable class of data, with bounded by possibly large total variations.

We then study the "slow erosion (or deposition) limit", obtained as the thickness of the moving layer tends to zero. We show that, in the limit, the profile of the standing layer depends only on the total mass of the avalanche flowing downhill, not on the time-law describing at which rate the material slides down. More precisely, the limiting slope of the mountain profile is provided by an entropy solution to a scalar integro-differential conservation law.

Same abstract as the talk.

We consider a model for the flow of granular matter which was proposed by Hadeler and Kuttler (Granular Matter, 1999). The original model uses the height of the standing layer and the thickness of the moving layer as the unknowns. By introducing the slope the standing layer, one arrives at a 2 by 2 system of balance laws. This system is weakly linearly degenerate at a point. With suitable conditions on the initial data, one can prove the global existence of smooth solutions. Furthermore, we prove the global existence of large BV solutions, for a class of initial data with bounded but possibly large total variation.

This is partly a joint work with Debora Amadori, Italy.

In this course we will give an introduction to conservative short capturing numerical methods for solving multi-dimensional systems of conservation laws. High order accurate finite difference, finite volume and discontinuous Galerkin finite element methods will be covered. We will start with the basic algorithm issues in a simple scalar one dimensional setting and then describe the generalization to multi-dimensional systems. A comparison among these different numerical methods will be provided.

Lecture References:

[1] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 1697, Springer, Berlin, 1998, pp.325-432.

[2] C.-W. Shu, Discontinuous Galerkin methods: general approach and stability, Numerical Solutions of Partial Differential Equations, S. Bertoluzza, S. Falletta, G. Russo and C.-W. Shu, Advanced Courses in Mathematics CRM Barcelona, Birkhäuser, Basel, 2009, pp.149-201.

The talk will be based on a joint work with L. Ambrosio, G. Crippa and A. Figalli. First, some new well-posedness results for continuity and transport equations with weakly differentiable velocity fields will be discussed. These results can be applied to the analysis of a 2 x 2 system of conservation laws in one space dimension known as the chromatography system, leading to global existence and uniqueness results for suitable classes of entropy admissible solutions.

We provide a bird's eye view of a selected topics in approximate solution of nonlinear conservation laws and related time dependent equations. We begin with a discussion on regularity spaces in theory and computation. We will continue with a presentation of the class of high-resolution central schemes. We will discuss the issue of entropy stability and conclude with ongoing research on constrained transport.

In this talk we discuss discretization based dispersive-dissipative regularization of mixed type systems and derive the resulting closure conditions known as kinetic relations. Algebraic kinetic relations link velocities of the undercompressed jump discontinuities with the corresponding driving forces and are widely used to model dynamical response of phase boundaries. To capture the effects of discretization more faithfully we propose to replace algebraic kinetic relations with differential kinetic equations which involve some specially selected collective variables characterizing not only the location of the discontinuity but also the structure of the transition region. Joint work with A. Vainchtein.

We consider a system of hyperbolic-parabolic equations describing a material instability mechanism associated to the formation of shear bands at high strain-rate plastic deformations of metals.

We consider the case of adiabatic shearing and derive a quantitative criterion for the onset of instability: Using ideas from the theory of relaxation systems we derive equations that describe the effective behavior of the system. The effective equation turns out to be a forward-backward parabolic equation regularized by fourth order term.

Further, we study numerically the effect of thermal diffusion on the evolution of these bands. It turns out that while localization initially forms at a later stage of the deformation heat diffusion has the power hinder and even altogether suppress localization and even return the evolution to uniform deformation. (joint work with Th. Katsaounis and Th. Baxevanis, Univ. of Crete).

In this talk, we present the study of the regularity of solutions to some systems of reaction–diffusion equations, with reaction terms having a subquadratic growth. We show the global boundedness and regularity of solutions, without smallness assumptions, in any dimension N. The proof is based on blow-up techniques. The natural entropy of the system plays a crucial role in the analysis. It allows us to use of De Giorgi type methods introduced for elliptic regularity with rough coefficients. Even if those systems are entropy supercritical, it is possible to control the hypothetical blow-ups, in the critical scaling, via a very weak norm.

We offer an alternate derivation for the symmetric-hyperbolic formulation of the equations of motion for a hyperelastic material with polyconvex stored energy. The derivation makes it clear that the expanded system is equivalent, for weak solutions, to the original system. We consider motions with variable as well as constant temperature. In addition, we present equivalent Eulerian equations of motion, which are also symmetric-hyperbolic.

Some recent research progresses on the multi-dimensional compressible Euler equations will be reviewed. In particular, the transonic flows past an obstacle such as an airfoil, and the isometric embeddings in geometry will be discussed. The talk is based on the joint works with Gui-Qiang Chen and Marshall Slemrod.

In this talk we study the stability of multidimensional contact discontinuities in compressible fluids. There are two kinds of contact discontinuities, one is so-called the vortex sheet, mainly due to that the tangential velocity is discontinuous across the front, and the other one is the entropy wave, for which the velocity is continuous while the entropy has certain jump on the front.

It is well-known that the vortex sheet in two dimensional compressible Euler equations is stable when the Mach number is larger than √2, while in three dimensional problem it is always unstable. But, some physical phenomena indicate that the magnetic field has certain stabilization effect for waves in fluids. The first goal of this talk is to rigorously justify this physical phenomenon, and to investigate the stability of three-dimensional current-vortex sheet in compressible magneto-hydrodynamics. By using energy method and the Nash-Moser iteration scheme, we obtain that the current-vortex sheet in three-dimensional compressible MHD is linearly and nonlinearly stable when the magnetic fields on both sides of the front are non-parallel to each other.

The second goal is to study the stability of entropy waves. By a simple computation, one can easily observe that the entropy wave is structurally unstable in gas dynamics. By carefully studying the effect of magnetic fields on entropy waves, we obtain that the entropy wave in three-dimensional compressible MHD is stable when the normal mag- netic field is continuous and non-zero on the front.

This is a joint work with Gui-Qiang Chen.

Joint work with Chiu-Yen Kao.

Buckley-Leverett (BL) equation arises in two-phase flow problem in porous media. It models the oil recovery by water-drive in one-dimension. Here we propose a central scheme for an extension of the BL equation which includes the dynamic effects in the pressure difference between the two phases and results in a third order mixed derivatives term in the modified BL equation. The numerical scheme is able to capture the admissible shocks which is the so-called nonclassical shock due to the violation of the Oleinik entropy condition.

The isentropic Euler equations form a system of conservation laws modeling compressible fluid flows with constant thermodynamical entropy. Due to the occurence of shock discontinuities, the total energy of the system is decreasing in time. We review the second order calculus on the wasserstein space of probability measures and show how the isentropic Euler equations can be interpreted as a steepest descent equation in this framework. We introduce a variational time discretization based on a sequence of minimization problems, and show that this approximation converges to a suitably defined measure-valued solution of the conservation law. Finally, we present some preliminary results about the numerical implementation of our time discretization.

We give a complete description of nonlinear waves and their pairwise interactions in isentropic gas dynamics. Our analysis includes rarefactions, compressions and shock waves. We describe the interaction in terms of a reference state and incident wave strengths, and give explicit estimates of the outgoing wave strengths. Our estimates are global in that they apply to waves of arbitrary strength, and they are uniform in the incoming reference state. In particular, the estimates continue to hold as this state approaches vacuum. We also consider composite interactions, which can be regarded as a degenerate superposition of pairwise interactions. We construct a class of exact weak solutions which demonstrate some interesting and surprising features of interactions, and use these to demonstrate the collapse of a vacuum: in most cases two shocks will emerge from the vacuum, but in certain asymmetric cases a single shock and a rarefaction may emerge.

Joint work with Blake Temple.

In this ongoing collaboration with Blake Temple, we attempt to prove the existence of periodic solutions to the Euler equations of gas dynamics. Such solutions have long been thought not to exist due to shock formation, and this is confirmed by the celebrated Glimm-Lax decay theory for 2×2 systems. However, in the full 3×3 system, multiple interaction effects can combine to slow down and prevent shock formation. We describe the physical mechanism supporting periodicity, analyze combinatorics of simple wave interactions, and develop periodic solutions to a "linearized" problem. These linearized solutions have a beautiful structure and exhibit several surprising and fascinating phenomena. We then consider perturbing these as a bifurcation problem, which leads us to problems of small divisors and KAM theory. As an intermediate step, we find solutions which are periodic to within arbitrarily high Fourier modes.

We introduce some results obtained by the author and his collaborators on existence, stability and uniqueness of transonic shocks and subsonic-supersonic flows in nozzles by using various different models.

We show that there are supersonic solutions to the Euler system that are not hyperbolic in the traditional sense. These solutions occur at the transonic region, whose characteristics may both come from the sonic line and end at the sonic line. Based on the new wave structure, we offer perspectives to construct global transonic solutions to the Riemann problems.

Course abstract: We examine from a classical dynamical systems point of view stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde.

Lecture 1 abstract: Stability of viscous shock waves. We discuss the basic types of viscous shock waves, the Evans function condition and its meaning, and outline a basic one-dimensional stability proof assuming that the Evans condition holds.

Using a combination of numerical Evans function computations and asymptotic ODE analysis, we carry out global stability analyses for interesting examples including ideal gas and parallel MHD shocks, across the entire range of physical parameters: in particular, in the large amplitude or magnetic field limit.

Elaborating on the basic stability theory, we examine conditional stability and Hopf bifurcation of possibly unstable viscous shock waves.

We discuss finally stability and bifurcation of flow in a channel with periodic boundary conditions: cellular bifurcation, pattern formation, and an Evans function construction for genuinely multi-dimensional (i.e., nonplanar) solutions.