CAREER: Thin shells - problems in nonlinear elasticity and fluid dynamics

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LewickaDMS-0846996 A unifying theme in this project is the asymptotic behaviorof given equations as the domain approaches a "degenerate" regionin the limit. The considered problems are nonlinear variationalor partial differential equations, and the degeneracy in questioncan take various forms: loss of dimension, loss of regularity, orunboundedness. The first set of questions relates to themathematical theory of nonlinear elasticity, which studies largemechanical deformations of three-dimensional elastic bodies. Theinvestigator undertakes a long-term research program, with thescope of: deriving lower-dimensional shell theories through themethods of Gamma-convergence and understanding their connectionswith the geometry of the mid-surface, analyzing the(infinitesimal) isometries of surfaces and the effects ofrigidity on the derived theories, studying nonlinear phenomenasuch as buckling and blistering for a given shell undercompression (one application is related to plant growth). Thesecond set of questions in this project relates to fluiddynamics. Boundary irregularities of various structures andscales are considered in the limit when the boundary behaviorbecomes degenerate. Other problems concern traveling fronts incombustion in unbounded channels, and the dynamics of solutionswith large initial data under the Navier boundary conditions inthin three-dimensional shells. Because elastic thin (or otherwise "degenerate") objects ofvarious geometries are ubiquitous in the physical world, theprecise understanding of laws governing their equilibria has manypotential applications. For example, many growing tissues(leaves, flowers, or marine invertebrates) exhibit complicatedconfigurations during their free growth and one would like toreproduce them with man-made means. A related long-standingproblem in the mathematical theory of elasticity is to rigorouslypredict theories of such lower-dimensional objects starting fromthe nonlinear theory of full three-dimensional objects. Forplates, a very recent effort has lead to rigorous justificationof a hierarchy of such theories, depending on the magnitude ofthe applied forces and resulting in stretching, crumpling,bending, or a combination of these. For shells (when themid-surface is curved), despite extensive use of their ad hocgeneralizations in the literature and engineering applications,much less is known from the mathematical point of view. Theinvestigator identifies several nonlinear problems in continuummechanics of solids and fluid dynamics, naturally posed inspecific degenerate domains, with the intention of rigorouslyunderstanding the behavior of the system based on generalprinciples. At the heart of the program are interestingconnections between calculus of variations, differentialequations, geometry, material science, fluid dynamics, numericalanalysis, and even biology. They have a potential to deliveruseful observations in e.g. structural mechanics, whileintegrating the goal of exposing scientific results to a broadercommunity at various education levels.