27 research outputs found
On admissibility criteria for weak solutions of the Euler equations
We consider solutions to the Cauchy problem for the incompressible Euler
equations satisfying several additional requirements, like the global and local
energy inequalities. Using some techniques introduced in an earlier paper we
show that, for some bounded compactly supported initial data, none of these
admissibility criteria singles out a unique weak solution.
As a byproduct we show bounded initial data for which admissible solutions to
the p-system of isentropic gas dynamics in Eulerian coordinates are not unique
in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof
Dissipative continuous Euler flows
We show the existence of continuous periodic solutions of the 3D
incompressible Euler equations which dissipate the total kinetic energy
Quasiconvexity at the boundary and the nucleation of austenite
Motivated by experimental observations of H. Seiner et al., we study the nucleation of austenite in a single crystal of a CuAlNi shape-memory alloy stabilized as a single variant of martensite. In the experiments the nucleation process was induced by localized heating and it was observed that, regardless of where the localized heating was applied, the nucleation points were always located at one of the corners of the sample - a rectangular parallelepiped in the austenite. Using a simplified nonlinear elasticity model, we propose an explanation for the location of the nucleation points by showing that the martensite is a local minimizer of the energy with respect to localized variations in the interior, on faces and edges of the sample, but not at some corners, where a localized microstructure, involving austenite and a simple laminate of martensite, can lower the energy. The result for the interior, faces and edges is established by showing that the free-energy function satisfies a set of quasiconvexity conditions at the stabilized variant in the interior, faces and edges, respectively, provided the specimen is suitably cut
Characterization of Generalized Young Measures Generated by Symmetric Gradients
This work establishes a characterization theorem for (generalized) Young measures generated by symmetric derivatives of functions of bounded deformation (BD) in the spirit of the classical Kinderlehrer\ue2\u80\u93Pedregal theorem. Our result places such Young measures in duality with symmetric-quasiconvex functions with linear growth. The \ue2\u80\u9clocal\ue2\u80\u9d proof strategy combines blow-up arguments with the singular structure theorem in BD (the analogue of Alberti\ue2\u80\u99s rank-one theorem in BV), which was recently proved by the authors. As an application of our characterization theorem we show how an atomic part in a BD-Young measure can be split off in generating sequences
Optimal Existence Theorems for Nonhomogeneous Differential Inclusions
AbstractIn this paper we address the question of solvability of the differential inclusions (1.1). Our approach to these problems is based on the idea of constructing a sequence of approximate solutions which converges strongly and makes use of Gromov's idea (following earlier work of Nash and Kuiper) to control convergence of the gradients by appropriate selection of the elements of the sequence. In this paper we identify an optimal setting of this method. In particular we show that the existence result holds for general upper semicontinuous functions H without extra requirements like quasiconvexity of H with respect to Du, which was assumed in previous works, where the idea to apply the Baire category lemma to the sets of approximate solutions was developed. We also apply our result to identify the minimal sets, where the function H should vanish to guarantee solvability of the inclusions
Comparing various methods of resolving nonconvex variational problems
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7 , Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
Minimizers of variational problems with Euler-Lagrange equations having measure-valued right-hand side
Consiglio Nazionale delle Ricerche - Biblioteca Centrale - P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal
A new approach to Young measure theory, relaxation and convergence in energy
Support by Russian Foundation for Basic Research, grant N 97-01-00508Consiglio Nazionale delle Ricerche - Biblioteca Centrale -. P.le Aldo Moro, 7, Rome / CNR - Consiglio Nazionale delle RichercheSIGLEITItal