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Register a new userFluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions
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We are concerned with underlying connections between fluids, elasticity, isometric embedding of Riemannian manifolds, and the existence of wrinkled solutions of the associated nonlinear partial differential equations. In this paper, we develop such connections for the case of two spatial dimensions, and demonstrate that the continuum mechanical equations can be mapped into a corresponding geometric framework and the inherent direct application of the theory of isometric embeddings and the Gauss-Codazzi equations through examples for the Euler equations for fluids and the Euler-Lagrange equations for elastic solids. These results show that the geometric theory provides an avenue for addressing the admissibility criteria for nonlinear conservation laws in continuum mechanics.
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In this article we prove the global existence of weak solutions for a diffuse interface model in a bounded domain (both in 2D and 3D) involving incompressible magnetic fluids with unmatched densities. The model couples the incompressible Navier-Stokes equations, gradient flow of the magnetization vector and the Cahn-Hilliard dynamics describing the partial mixing of two fluids. The density of the mixture depends on an order parameter and the modelling, specifically the density dependence, is inspired from Abels, Garcke and Gr{u}n 2011.
Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from a Riemannian manifold to a sphere (in the spirit of Friesecke--James--M{u}ller [20]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy--Green tensors and second fundamental forms, which extends the Ciarlet--Mardare theorem in [17] to arbitrary dimensions and co-dimensions.
We consider wave maps on $(1+d)$-dimensional Minkowski space. For each dimension $dgeq 8$ we construct a negatively curved, $d$-dimensional target manifold that allows for the existence of a self-similar wave map which provides a stable blowup mechanism for the corresponding Cauchy problem.
We construct global weak solutions to isothermal quantum Navier-Stokes equations, with or without Korteweg term, in the whole space of dimension at most three. Instead of working on the initial set of unknown functions, we consider an equivalent reformulation, based on a time-dependent rescaling, that we introduced in a previous paper to study the large time behavior, and which provides suitable a priori estimates, as opposed to the initial formulation where the potential energy is not signed. We proceed by working on tori whose size eventually becomes infinite. On each fixed torus, we consider the equations in the presence of drag force terms. Such equations are solved by regularization, and the limit where the drag force terms vanish is treated by resuming the notion of renormalized solution developed by I. Lacroix-Violet and A. Vasseur. We also establish global existence of weak solutions for the isothermal Korteweg equation (no viscosity), when initial data are well-prepared, in the sense that they stem from a Madelung transform.
We show existence of solutions for the equations of static atomistic nonlinear elasticity theory on a bounded domain with prescribed boundary values. We also show their convergence to the solutions of continuum nonlinear elasticity theory, with energy density given by the Cauchy-Born rule, as the interatomic distances tend to zero. These results hold for small data close to a stable lattice for general finite range interaction potentials. We also discuss the notion of stability in detail.
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