Weak Convergence Methods for Nonlinear Partial Differential Equations

Evans, Lawrence C.

doi:10.1090/cbms/074

Cited by 572 publications

(370 citation statements)

References 0 publications

Supporting

Mentioning

368

Contrasting

Unclassified

Order By: Relevance

“…Hence it follows that there exists a unique solution to (Q r ). In addition, we have from (2.5) and (2.2b) that 14) and hence the bound on u r in (2.9a). Finally, it follows from (2.2b) that…”

Section: A)mentioning

confidence: 91%

A Mixed Formulation of the Monge-Kantorovich Equations

Barrett¹,

Prigozhin

2007

ESAIM: M2AN

View full text Add to dashboard Cite

Abstract. We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart-Thomas element. Finally, we present some numerical experiments, where both the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems involving obstacles and regions of cheap transportation.

show abstract

Section: A)mentioning

confidence: 91%

A Mixed Formulation of the Monge-Kantorovich Equations

Barrett¹,

Prigozhin

2007

ESAIM: M2AN

View full text Add to dashboard Cite

show abstract

“…The limit ^^, ^ t oo for fixed h > 0 will dramatically simplify the "energy landscape". We believe that this procedure is essential in capturing the "generic" behaviour of the gradient flux in the regime (14). This procedure also has the more technical advantage that the natural implicit discretization of the gradient flux gives us a sequence of variational problems, whose behaviour for ^^,^ t °o can be analyzed within the framework of F-<:onvergence.…”

Section: A Dynamic Selection Principlementioning

confidence: 99%

Dynamics of Labyrinthine Pattern Formation in Magnetic Fluids: A Mean-Field Theory

Otto

1998

Archive for Rational Mechanics and Analysis

117

View full text Add to dashboard Cite

We are interested in the flow of a droplet of viscous ferrofluid in the Hele-Shaw cell under a transverse magnetic field. The (twodimensional) phase configuration is observed to evolve into a labyrinthine pattern. We show that the conventional model for this flow has the form of a gradient flux w.r.t. an energy functional, which is the sum of magnetic and surface energy. In particular, we are interested in the behaviour of this flow problem in the regime of large magnetization M 2 » 1. In this regime, the details of the pattern evolution are observed to be highly sensitive to changes in the initial configuration. This is reflected in the linear stability analysis of the circular phase configuration, a stationary point of the dynamics which is more and more unstable in the limit M 2 f oo. In order to capture the "generic" behaviour of the dynamical system in this regime, we need a selection principle which dismisses those non-generic solutions. We propose a selection principle for the limit M 2 t oo which is based on the natural implicit discretization in time of our gradient flux formulation. We prove that this approach leads (in an appropriate scaling) to the equation dts-As 2 = 0 for s(t, x) 6 [0,1], the local spatial average of the phase configuration X(<,s) € {0,1} (x(t,x) = 1 if x € R 2 lies in the two-dimensional cross-section of the fluid at time t, xfa x ) -0 e^e )-Thus this quantity, which contains information on the "microstructured zone", evolves detenninistically, although x k essentially unpredictable.

show abstract

“…Our real intention is rather to try to gain some further insight into the analytical behavior and physical significance of the nonlinear term Vi v 2 and its rotated counterpart (v 1)2 -(v 2 )2 • A vague tenet of compensated compactness theory (cf. [7,Chapter 5]) holds that "physically natural" nonlinear terms in PDE should have better analytic properties than similar, but nonphysical, terms. We surely have here an instance of this phenomenon but are also currently lacking any really deep insight.…”

Section: Introductionmentioning

confidence: 99%