Wednesdays at 4 PM in Eckhart 207A.
We study the inviscid limit of solutions of the evolutionary Navier-Stokes system in a two space dimensional bounded domain with nonhomogeneous boundary conditions admitting flow across the boundary. Under a geometrical restriction on the shape of the domain we obtain a suitable bound on the $L_\infty$-norm of the vorticity, which is independent of the viscosity and also uniform in space. This estimate guarantees the existence of the limit of solutions as the viscosity tends to zero, which satisfies the Euler system for incompressible fluids. The result is applicable in presence of the flux through the boundary of arbitrary strength.
I will describe the construction of local minimizers to the Ginzburg-Landau energy in certain 3-d domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment.
An extremely attractive but very difficult problem of fluid dynamics is to find a regular way of constructing functionals which grow monotonically with time. Existence of such a functional would mean that an initially `small' solution gradually becomes `large', which is closely related to such fundamental problems as instability of fluid flows, magnetohydrodynamic dynamo, etc. One such functional known for a long time, it can be called `virial'. First it appeared in classical mechanics in the `virial equation'. It was first introduced in the 'virial theorem' and then extensively used in proving of instability in finite-dimensional mechanical systems. Then the `virial equality' had been used in problems of fluid equilibria instability in astrophysical applications and for fluid possessing a free surface. At the same time, the rational reason, why monotonically increasing `virial' functions or functionals should exist, is unknown (and if they do exist what does define their mathematical structure). In the present paper, we discuss one possible general way of introduction the `virial functional' in fluid dynamics from a universal viewpoint. Our approach is based on the equalities underlying the Hamilton principle of least action. We can introduce the notion of `virial' (and show that it is useful) for a fluid system as soon as we know the expression for its Lagrangian. Technically, we consider families of vector field depending on a scalar parameter. The variations of velocities (and other unknown fields) correspond to derivatives with respect to that parameter. The `virial' appears in the equality for the second derivative of Lagrangian with respect to that parameter (or in other words in the equation for second variation of the action functional). It shows the close link to the solution of the well-known `Jacobi equation for geodesic deviations'. We use the `virial' to obtain results concerning nonlinear and linear {\it a priori} estimates of solutions. Particular cases of an inviscid incompressible fluid (with and without free surface), stratified fluid, and ideal magnetohydrodynamics have been considered. Our approach recovers and unifies all known instability results obtained using the `virial'. In particular, it shows that the the `horseshoe' instability and instability considered earlier by Arnold can be discovered and studied using the direct Lyapunov method with the `virial' as the Lyapunov functional.
We begin with some background on the physical behavior of thin films: how films are synthesized and patterned, how deformation may occur in films of so-called active materials. We then present the basic intuition that underlies plate and shell theories. This concerns the relation between thinness and the geometry of SO(3) and a discussion of the scales at which ``membrane'' and ``bending'' theories emerge. We then examine the rigorous direct passage from 3-D elasticity to thin film theory with and without interfacial energy. Problems of this kind have been open for some 250 years, but recent multiscale mathematical methods - essentially, a better understanding of the relation between weak and strong convergence in the presence of certain soft differential constraints - have clarified (rigorously, without an ansatz) the status of many of the plate and shell theories that appear in the literature. We discuss first the membrane theory. In the case that the membrane theory is trivial, we discuss the derivation of the bending energy. The latter relies on a quantitative Reshetnyak-Liouville theorem (joint work with Friesecke and Mueller), that gives an estimate of the deviation of a function from linearity in terms of the deviation of its gradient from SO(n). We then highlight some predictions that are unique to thin films of active materials, in particular, the presence of interfaces that are possible in films but not in the corresponding bulk material (joint work with Bhattacharya). We suggest how the presence of these interfaces might be exploited to create motion at small scales using tents, tunnels, wedges, pacmen. We conclude the lecture with a look at a naturally occurring virus, Bacteriophage T-4, whose tail undergoes a kind of phase transformation that is apparently much like that predicted for the films.
Systems of conservations laws models a wide range of physical phenomena. For one dimensional problems there is a fairly well developed theory for small solutions. We will consider examples where blowup behavior occurs, showing that the restriction to small data can be essential. We will also consider the situation for physical systems such as compressible Navier-Stokes in several space dimensions.
Flame propagation, when initalized at a source point, is in general anything but spherical. Evidences of such a propagation mode were, however, dicovered by P. Ronney in experiments dating back to the early 90's. A mechanism for the propagation of spherical flames is proposed in an important paper of G. Joulin (Comb. Sci. Tech, 1985). In this work, a slow time scale - proportional to the square of the normalized activation energy - is identified. The flame radius is then found, through formal matched asymptotic expansions, to satisfy a nonlinear integro-differential equation. An interesting feature of Joulin's analysis is that it can be carried over to more complex models: models with heat losses, advection by a turbulent flow, presence of dust... It seems therefore relevant to ask for a mathematically rigorous derivation of Joulin's model. The goal of this talk is to present the mathematical arguments that make this derivation work.
Several Glimm-type functionals for approximate solutions of nonlinear hyperbolic systems of conservation laws have been introduced in recent years. In this talk, we provide a general framework to prove that such functionals can be extended to general functions with bounded variation and are lower semi-continuous in the strong $L^1$ topology. In particular, our analysis covers systems with general flux-functions. An an illustration of the use of the continuous Glimm-type fucntionals, we study the qualitative behavior of solutions of hyperbolic systems with general (non-genuinely nonlinear) characteristic fields.
We present some joint work with Etienne Sandier, about the complex-valued Ginzburg-Landau functional. First we give some estimate on the "vorticity" associated to Ginzburg-Landau configurations, which translates in the time-dependent case into a sharp upper bound on the velocities of vortices (in any dimension). Secondly, we turn to the heat-flow Ginzburg-Landau equation and present a scheme which allows to use such an estimate to deduce the limiting dynamical law of vortex-motion.
We present a two-scale framework for approximating the solution of a second order elliptic problem in divergence form. It results in a coarse-scale problem (the "upscaled problem") coupled to a series of localized problems. We then devise an efficient and mostly parallelizable two-scale, subgrid approximation. Its solution requires solving a series of small, fine-grid local problems (sub-grid problems), and a single global coarse-grid problem. The method uses any order mixed finite element spaces. An important feature of the method is that it maintains the principle of conservation on the fine scale. Analysis of the method shows optimal (in a two-scale sense) error bounds independent of the two-scale decomposition. A special choice of spaces provides a practical and accurate method. Application to subsurface flow is discussed to illustrate the effectiveness and applicability of the method. We also consider briefly the application of the two-scale approximation as a preconditioner for a direct approximation of the full problem.
We study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically (with d>1 incommensurate frequencies). We show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d-dimensional torus that preserves a foliation by irrational straight lines. We explain how some dynamical features translate into properties for the field in the cavity. In particular, we show that when the torus map satisfies a KAM theorem -- which happens for a Cantor set of positive measure of parameters -- the energy of the electromagnetic field remains bounded. When the torus map is in a resonant region -- which happens in open sets of parameters inside the gaps of the previous Cantor set -- the energy grows exponentially.
The talk will be devoted to the stability of conical-shaped solutions of a class of reaction-diffusion equations in dimension 2. Such equations arise especially in combustion models. A characterization of the global solutions under some conditions at infinity, as well as the linear and global stability of the fronts will be discussed. This talk is based on a joint work with R. Monneau (ENPC) and J.-M. Roquejoffre (Toulouse).
A crystal is a periodic lattice of atoms. Under deformation a crystalline lattice may change, and undergo what is known as a solid-to-solid martensitic transformation. A polycrystal is a composite of single crystalline grains, each one of whose crystalline lattices is oriented in some fashion. The deformation-induced martensitic transformation of a polycrystal is governed by overall elastic misfit energy minimization, and, thus, the martensitic transformation of a single grain depends on the martensitic transformations of all other grains. In this talk we explore this dependence on the example of a two-dimensional laminated polycrystal.
A framework for understanding the global structure of near integrable $n$ d.o.f. systems is proposed. The goal is to reach a similar situation to the forced 1 d.o.f. systems, where one is able in a glance of the integrable phase portrait, understand where instabilities are expected to arise. This is achieved for some types of Hamiltonians by plotting the energy-momentum bifurcation diagram and the branched surfaces - our generalization of Fomenko graphs to higher dimensions. Thus a classification of such systems is found and the instabilities associated with the typical features of such systems are demonstrated numerically. Furthermore, using these tools we prove that the existence of a strongly resonant lower dimensional torus of certain kinds on an energy surface implies that the energy surfaces change their topology at this energy level. Thus a new connection between dynamics and topology is found. Joint work with A. Litvak-Hinnenzon.
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