Nonlinear PDEs: Theory, Numerics, and Applications
Oslo, April 26-27, 2004

ABSTRACTS (chronologically ordered):

François Bouchut
I shall explain the different approaches, with the recent developments, to prove uniqueness and weak stability of solutions to linear transport equations with discontinuous coefficient in several dimensions, with an emphasis on what is expected depending on the assumptions on the coefficient.

References

• L. Ambrosio, transport equations and Cauchy problem for BV vector fields, to appear in Invent. Math.
• F. Bouchut, F. James, S. Mancini, Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient, preprint 2004.

Sigmund Selberg
I will describe a recent result, obtained in collaboration with N. J. Mauser, on the joint nonrelativistic (light speed tends to infinity) and semiclassical limit (Planck constant tends to 0) of the Dirac-Maxwell system, which is the fundamental equation in quantum electrodynamics.

We prove that, under suitable assumptions on the initial state, the solution exists in a uniform time interval, and a subsequence of the Wigner transform of the quantum state operator converges weakly (on this time interval) to a solution of the Vlasov-Poisson system.

Nils Henrik Risebro
We consider numerical schemes for the Hunter--Saxon equation
$$v_t + uv_x = -\frac{1}{2}v^2,\qquad u_x=v.$$

This is a model equation describing some features for the director field of a nematic liquid crystal. Formally we have that
$$\left( u_t + u u_x\right)_x = \frac{1}{2}\int_0^x \left(u_x(y,t)\right) 2\,dy,$$

so we see that this equation has some similarity to Burgers'
equation with a source term. However it turns out that in contrast to Burgers' equation, solutions of the Hunter--Saxon equation do not have shocks, but only ``cusps'', i.e., isolated points where
$u_x=-\infty$, and $u$

itself is continuous at these points.

We examine three schemes for this equation, and show that they produce a convergent subsequence of H\"older continuous approximations by establishing uniform Sobolev estimates.

Achim Schroll
Classical finite volume methods simulating conservation laws are of low (first) order of accuracy. However to compute more complex flows featuring wave interactions especially in multi dimensions higher order methods are needed. The state-of the-art approach to increase the order of the methods is to reconstruct a profile of the state variables based on the available cell averages. Classical approximation theory suggests polynomial reconstruction as applied in WENO schemes. However flows featuring slip lines are very sensitive to small perturbations (vortex formation, Kelvin Helmholtz instability) and (W)ENO schemes require an expensive selection mechanism the control the variation of the polynomial ansatz functions.

Only recently non-polynomial reconstruction techniques have been developed. The goal is to find non-oscillatory ansatz functions which provide enough freedom (parameters) to archive high accuracy and at the same time minimize the variation: High accuracy at low variation!

In this talk third and fifth order hyperbolic and logarithmic reconstruction will be discussed. The highlight is a compact, limiter-free, double logarithmic reconstruction of third order of accuracy everywhere, even near shocks and at local extrema!

Applications include one- and two component gas flows in two dimensions: Shock/bubble interactions, vortex and turbulence formation. Also peakon interactions governed by the Camassa-Holm equation and simulated by an adaptive FV scheme will be presented.

Siddhartha Mishra
In this talk, we present a new algorithm to restore noisy images. The algorithm is based on the solution of a linear elliptic equation with variable coefficients. The coefficient is choosen by using the topological gradient. The level sets of the coefficient serve as a good starting point for a simple algorithm to segment the image. The results of the algorithm on some test images is presented.

Snorre Christiansen
For problems in electromagnetics the Edge Elements of Nedelec have emerged as the spaces of choice for discretizations : Contrary to other so-called nodal finite element spaces they have good approximation properties in the presence of reentrant corners and they do not yield spurious eigenvalues. In this talk I will show that these spaces also have a nice property relevant to convergence theory for nonlinear PDEs. A variant of Murat and Tartar's Div Curl lemma is proved in a setting where the divergence is controlled only weakly with a Galerkin method.

Corrado Mascia
Hyperbolic systems with relaxation support smooth traveling waves corresponding to discontinuous shock waves for appropriate systems of conservation laws. These are usually called ``relaxed'' system and can be seen as the singular limit of the original relaxation system. The traveling waves of the relaxation system are expected to inherit stability from the corresponding shock of the relaxed system.

Under the weak assumption of spectral stability, or stable point spectrum of the linearized operator about the wave, in ajoint work with K.Zumbrun (Indiana University) we estabilish sharp pointwise Green's function bounds and consequent linear and nonlinear orbital stability. A consequence is stability of small-amplitude profiles of Broadwell and Jin-Xin models for easch of which spectral stability has been verified in other works.

Xue-Cheng Tai
The level set methods are power tools in tracing interfaces for different applications. In this work, we will present some variants of the level set methods and use them for image segmentation, electrical impedance tomography (EIP) and Positron Emission tomography.

In classical level set approaches, the sign of $n$ level set functions are utilized to identify up to $2^n$ phases. The novelty in our approach is to introduce a piecewise constant level set function and use each constant value to represent a unique phase. If $2^n$ phases should be identified, the level set function must approach $2^n$ predetermined constants. We just need one level set function to represent an arbitrary number of phases and this gains the storage capacity.
Further, the reinitializing procedure requested in classical level set methods is superfluous using our approach. The minimization functional for our approach is convex and differentiable and thus avoid some of the problems with the non-differentiability of the Delta and Heaviside functions. Numerical examples will be given and we shall also compare our method with related approaches.

References:

1. Tony F. Chan and Xue-Cheng Tai: Level Set And Total Variation Regularization For Elliptic Inverse Problems With Discontinuous Coefficients, Journal of Computational Physics, Vol. 193 (2003), pp. 40-66.
2. Johan Lie, Marius Lysaker and Xue-Cheng Tai, A Variant of the Level Set Method and Applications to Image Segmentation, September 2003. UCLA, Applied Mathematics, CAM-report-03-50.
3. Eric T. Chung, Tony F. Chan and Xue-Cheng Tai, Electrical Impedance Tomography Using Level Set Representation and Total Variational Regularization, November 2003, UCLA, Applied Mathematics, CAM-report-03-64.

Ragnar Winther
There have been many attempts during the past four decades to construct stable mixed finite elements for the Hellinger-Reissner formulation of linear elasticity, i.e., the stress-displacement formulation. Unfortunately, these efforts have not been as successful as expected. There is now renewed interest in this topic due to applications of mixed models in areas such as viscoelasticity, where the stress strain relation may be nonlocal, and, as a consequence, a pure displacement model is excluded. In this talk, we first explain why the condition of symmetry of the stress tensor makes it difficult to construct stable, low order, mixed finite elements for elasticity. By introducing a proper commuting diagram, we establish a connection between the standard de Rham sequence and a corresponding "elasticity sequence." Utilizing discrete versions of this connection, we are then able to construct new stable elements in two and three space dimensions, which satisfy either the usual symmetry condition or a weak version of this condition.

This represents joint work with Douglas N. Arnold, Univ. of Minnesota and Richard S. Falk, Rutgers University.

Raimund Bürger
This talk is divided into two parts. In the first part, we formulate and partly analyze a new mathematical model for continuous sedimentation-consolidation processes of flocculated suspensions in so-called clarifier-thickener units. This model appears in two variants for cylindrical and variable cross-sectional area units, respectively (Models 1 and 2). In both cases, the governing equation is a scalar, strongly degenerate parabolic equation in which both the convective and diffusion fluxes depend on parameters that are discontinuous functions of the depth variable. The initial-value problem for this equation is analyzed for Model 1. We introduce a simple finite-difference scheme and prove its convergence to a weak solution that satisfies an entropy condition. A limited analysis of steady states as desired stationary modes of operation is performed. Numerical examples illustrate that the model realistically describes the dynamics of flocculated suspensions in clarifier-thickeners.

In the second part, a very similar mathematical framework is applied to a model of traffic flow. The well-known Lighthill Whitham-Richards kinematic traffic flow model for unidirectional flow on a single-lane highway is extended to include both abruptly changing road surface conditions and drivers' reaction time and anticipation length. The result is again a strongly degenerate convection-diffusion equation, where the diffusion term, accounting for the drivers' behaviour, is effective only where the local car density exceeds a critical value, and the convective flux function depends discontinuously on the location. It is shown that the validity of the proposed traffic model is also supported by a recent mathematical well-posedness (existence and uniqueness) theory for quasilinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. This theory includes a convergence proof for a monotone finite-difference scheme, which is used herein to simulate the traffic flow model for a variety of situations.

This talk is based on joint work with Kenneth H. Karlsen and John D. Towers.

Peter Lindquist
This is a joint work with P. Juutinen.

A classical theorem of Rad\'{o} states that, if a continuous function $f(z)$ is holomorphic in the open set where $f(z)\neq 0$, then it is holomorphic in the whole domain. The level set $f(z)=0$ has been "removed''. The same phenomenon occurs for the Laplace equation: {\it Suppose that $u$ is of class $C^1(\Omega)$. If $u$ is harmonic in the subset where $u(x)\neq 0$, then $u$ is harmonic in the whole $\Omega$.} The theorem has been generalized to linear elliptic and parabolic equations by Shabat and Kral.

We study the removability of a level set for rather general elliptic and parabolic quasilinear equations of the second order. For example, the minimal surface equation, the $p$ Laplace equation, the Burgers equation, and the heat equation are included. We work directly with {\it viscosity solutions}. Our method is based on a device to treat a critical point: if $\nabla u(x_0)=0$ then no testing at all is needed at that point. This relaxation is based on a rather heavy machinery in the theory of viscosity solutions.

Reference:
www.maths.jyu.fi/~peanju/preprints/

Espen R. Jakobsen
I will discuss resent results on error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations. These are second order degenerate elliptic and fully non-linear equations having non-smooth solutions. They appear in optimal stochastic control theory which has many applications e.g. in finance. For more than a decade, nobody was was able to obtain error bounds for numerical schemes for such equations. The breakthrough was made by Krylov in 1997 and 2000, and more recently these results have been improved and generalized by Barles and the speaker.