Marta Lewicka

The Foppl-von Karman equations for plates with incompatible strains

We provide a derivation of the Föppl-von Kármán equations for the shape of and stresses in an elastic plate with incompatible or residual strains. These might arise from a range of causes: inhomogeneous growth, plastic deformation, swelling or shrinkage driven by solvent absorption. Our analysis gives rigorous bounds on the convergence of the three-dimensional equations of elasticity to the low-dimensional description embodied in the plate-like description of laminae and thus justifies a recent formulation of the problem to the shape of growing leaves. It also formalizes a procedure that can be used to derive other low-dimensional descriptions of active materials with complex non-Euclidean geometries.

The uniform Korn-Poincaré inequality in thin domains

Abstract We study the Korn–Poincare inequality: ‖ u ‖ W 1 , 2 ( S h ) ⩽ C h ‖ D ( u ) ‖ L 2 ( S h ) , in domains S h that are shells of small thickness of order h, around an arbitrary compact, boundaryless and smooth hypersurface S in R n . By D ( u ) we denote the symmetric part of the gradient ∇u, and we assume the tangential boundary conditions: u ⋅ n → h = 0 on ∂ S h . We prove that C h remains uniformly bounded as h → 0 , for vector fields u in any family of cones (with angle π / 2 , uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We show that this condition is optimal, as in turn every Killing field admits a family of extensions u h , for which the ratio ‖ u h ‖ W 1 , 2 ( S h ) / ‖ D ( u h ) ‖ L 2 ( S h ) blows up as h → 0 , even if the domains S h are not rotationally symmetric.

Models for elastic shells with incompatible strains

The three-dimensional shapes of thin lamina, such as leaves, flowers, feathers, wings, etc., are driven by the differential strain induced by the relative growth. The growth takes place through variations in the Riemannian metric given on the thin sheet as a function of location in the central plane and also across its thickness. The shape is then a consequence of elastic energy minimization on the frustrated geometrical object. Here, we provide a rigorous derivation of the asymptotic theories for shapes of residually strained thin lamina with non-trivial curvatures, i.e. growing elastic shells in both the weakly and strongly curved regimes, generalizing earlier results for the growth of nominally flat plates. The different theories are distinguished by the scaling of the mid-surface curvature relative to the inverse thickness and growth strain, and also allow us to generalize the classical Föppl–von Kármán energy to theories of prestrained shallow shells.

The Infinite Hierarchy of Elastic Shell Models: Some Recent Results and a Conjecture

We summarize some recent results of the authors and their collaborators, regarding the derivation of thin elastic shell models (for shells with mid-surface of arbitrary geometry) from the variational theory of 3d nonlinear elasticity. We also formulate a conjecture on the form and validity of infinitely many limiting 2d models, each corresponding to its proper scaling range of the body forces in terms of the shell thickness.

Infinitesimal Isometries on Developable Surfaces and Asymptotic Theories for Thin Developable Shells

We perform a detailed analysis of first order Sobolev-regular infinitesimal isometries on developable surfaces without affine regions. We prove that given enough regularity of the surface, any first order infinitesimal isometry can be matched to an infinitesimal isometry of an arbitrarily high order. We discuss the implications of this result for the elasticity of thin developable shells.

Stability Conditions for Patterns of Noninteracting Large Shock Waves

In this paper we study different conditions whose presence is required for [A.] the admissibility and stability of large shocks present in solutions of a strictly hyperbolic n × n system of conservation laws in one space dimension ut + f(u)x = 0, [B.] the solvability and L1 well posedness of the Cauchy problem for the above equation, near solutions containing large and stable, but noninteracting shock waves. We compare the corresponding conditions of type A and B appearing in the literature; in particular, we show that the finiteness and stability conditions used in our most recent works generalize and/or unify these conditions in appropriate ways.

Travelling waves in two-dimensional reactive Boussinesq systems with no-slip boundary conditions

We consider systems of reactive Boussinesq equations in two-dimensional strips that are not aligned with gravitys direction. We prove that for any width of such strips and for arbitrary Rayleigh and Prandtl numbers, the systems admit smooth, non-planar travelling wave solutions with the fluids velocity satisfying no-slip boundary conditions.

Nonlinear conservation laws and applications

Foreword.- Preface.- Open questions in the theory of one dimensional hyperbolic conservation laws.- Multidimensional conservation laws: Overview, problems, and perspective.- Mathematical analysis of fluids in motion.- Selected topics in approximate solutions of nonlinear conservation laws.- High-resolution central schemes.- Stability and dynamics of viscous shock waves.- Mathematical aspects of a model for granular flow.- The flow associated to weakly differentiable vector fields: recent results and open problems.- Existence and uniqueness results for the continuity equation and applications to the chromatography system.- Finite energy weak solutions to the quantum hydrodynamics system.- The Monge problem in geodesic spaces.- Existence of a unique solution to a nonlinear moving-boundary problem of mixed type arising in modeling blood flow.- Transonic flows and isometric embeddings.- Well posedness and control in models based on conservation laws.-Homogenization of nonlinear partial differential equations in the context of ergodic algebras: Recent results and open problems.- Conservation laws at a node.- Nonlinear hyperbolic surface waves.- Vacuum in gas and fluid dynamics.- On radially symmetric solutions to conservation laws.- Charge transport in an incompressible fluid: New devices in computational electronics.- Localization and shear bands in high strain-rate plasticity.- Hyperbolic conservation laws on spacetimes.- Reduced theories in nonlinear elasticity.- Mathematical, physical and numerical principles essential for models of turbulent mixing.- On the Euler-Poisson equations of self-gravitating compressible fluids.- Viscous system of conservation laws: Singular limits.- A two-dimensional Riemann problem for scalar conservation laws.- Semi-hyperbolic waves in two-dimensional compressible Euler systems.- List of summer program participants.

Existence and Stability of Viscoelastic Shock Profiles

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic–parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and nonclassical type shock profiles.

On the Existence of Traveling Waves in the 3D Boussinesq System

We extend earlier work on traveling waves in premixed flames in a gravitationally stratified medium, subject to the Boussinesq approximation. For three-dimensional channels not aligned with the gravity direction and under the Dirichlet boundary conditions in the fluid velocity, it is shown that a non-planar traveling wave, corresponding to a non-zero reaction, exists, under an explicit condition relating the geometry of the crossection of the channel to the magnitude of the Prandtl and Rayleigh numbers, or when the advection term in the flow equations is neglected.