arXiv subjects
Publications and source records attributed to Marshall Slemrod.
We formulate new admissibility criteria for initial value problems motivated by the least action principle. These are applied to a two-dimensional Riemann initial value problem for the isentropic compressible Euler fluid flow. It is shown that the criterion prefers the 2-shock solution to solutions obtained by convex integration by Chiodaroli and Kreml or to the hybrid solutions recently constructed by Markfelder and Pellhammer.
A vector field similar to those separately introduced by Artstein and Dafermos is constructed from the tangent to a monotone increasing one-parameter family of non-concentric circles that touch at the common point of intersection taken as the origin. The circles define and space-fill a lens shaped region $\Omega$ whose outer and inner boundaries are the greatest and least circles. The double cusp at the origin creates a geometric singularity at which the vector field is indeterminate and has non-unique limiting behaviour. A semi-inverse method that involves the Airy stress function then shows that the vector field corresponds to the displacement vector field for a linear plane compressible non-homogeneous isotropic elastostatic equilibrium problem in $\Omega$ whose boundaries are rigidly rotated relative to each other, possibly causing rupture or tearing at the origin. A sequence of solutions is found for which not only are the Lam\'{e} parameters strongly-elliptic, but the non-unique limiting behaviour of the displacement is preserved. Other properties of the vector field are also established.
This paper provides a new admissibility criterion for choosing physically relevant weak solutions of the equations of Lagrangian and continuum mechanics when non-uniqueness of solutions to the initial value problem occurs. The criterion is motivated by the classical least action principle but is now applied to initial value problems which exhibit non-unique solutions. Examples are provided to Lagrangian mechanics and the Euler equations of barotropic fluid mechanics. In particular, we show the least action admissibility principle prefers the classical two shock solution to the Riemann initial value problem to certain solutions generated by convex integration. On the other hand, Dafermos's entropy criterion prefers convex integration solutions to the two shock solutions. Furthermore, when the pressure is given by $p(\rho)=\rho^2$, we show that the two shock solution is always preferred whenever the convex integration solutions are defined for the same initial data.
Examples of dynamical systems proposed by Z. Artstein and C. M. Dafermos admit non-unique solutions that track a one parameter family of closed circular orbits contiguous at a single point. Switching between orbits at this single point produces an infinite number of solutions with the same initial data. Dafermos appeals to a maximal entropy rate criterion to recover uniqueness. These results are here interpreted as non-unique Lagrange trajectories on a particular spatial region. The corresponding velocity is proved consistent with plane steady compressible fluid flows that for specified pressure and mass density satisfy not only the Euler equations but also the Navier-Stokes equations for specially chosen volume and (positive) shear viscosities. The maximal entropy rate criterion recovers uniqueness.
This paper examines a system of partial differential equations describing dislocation dynamics in a crystalline solid. In particular we consider dynamics linearized about a state of zero stress and use linear semigroup theory to establish existence, uniqueness, and time asymptotic behavior of the linear system.
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 Riemannian manifolds to spheres (in the spirit of Friesecke-James-M\"{u}ller [23]), 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 [18] to arbitrary dimensions and co-dimensions.
Direct linkages between regular or irregular isometric embeddings of surfaces and steady compressible or incompressible fluid dynamics are investigated in this paper. For a surface $(M,g)$ isometrically embedded in $\mathbb{R}^3$, we construct a mapping which sends the second fundamental form of the embedding to the density, velocity, and pressure of steady fluid flows on $(M,g)$. From the PDE perspectives, this mapping sends solutions to the Gauss--Codazzi equations to the steady Euler equations. Several families of special solutions of physical or geometrical significance are studied in detail, including the Chaplygin gas on standard and flat tori, as well as the irregular isometric embeddings of the flat torus. We also discuss tentative extensions to multi-dimensions.
We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings.
A theory for the evolution of a metric $g$ driven by the equations of three-dimensional continuum mechanics is developed. This metric in turn allows for the local existence of an evolving three-dimensional Riemannian manifold immersed in the six-dimensional Euclidean space. The Nash-Kuiper theorem is then applied to this Riemannian manifold to produce a wild evolving $C^{1}$ manifold. The theory is applied to the incompressible Euler and Navier-Stokes equations. One practical outcome of the theory is a computation of critical profile initial data for what may be interpreted as the onset of turbulence for the classical incompressible Navier-Stokes equations.
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.
In this paper the method of compensated compactness is applied to the problem of isometric immersion of a two dimensional Riemannian manifold with negative Gauss curvature into three dimensional Euclidean space. Previous applications of the method to this problem have required decay of order $t^{-4}$ in the Gauss curvature. Here we show that the decay of Hong $t^{-2-δ/2}$ where $δ\in(0,4)$ suffices.
The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.
The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A natural split to fast and slow state variables is, however, not available. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolving observables; averaging equations for the latter lead to computation of characteristic features of the motion. Such computations are presented in the paper.
We establish the weak continuity of the Gauss-Coddazi-Ricci system for isometric embedding with respect to the uniform $L^p$-bounded solution sequence for $p>2$, which implies that the weak limit of the isometric embeddings of the manifold is still an isometric embedding. More generally, we establish a compensated compactness framework for the Gauss-Codazzi-Ricci system in differential geometry. That is, given any sequence of approximate solutions to this system which is uniformly bounded in $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), then the approximating sequence has a weakly convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci system. Furthermore, a minimizing problem is proposed as a selection criterion. For these, no restriction on the Riemann curvature tensor is made.
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as initial and/or boundary value problems for a system of nonlinear partial differential equations of mixed elliptic-hyperbolic type whose mathematical theory is largely incomplete. In this paper, we develop a general approach, which combines a fluid dynamic formulation of balance laws for the Gauss-Codazzi system with a compensated compactness framework, to deal with the initial and/or boundary value problems for isometric immersions in $\R^3$. The compensated compactness framework formed here is a natural formulation to ensure the weak continuity of the Gauss-Codazzi system for approximate solutions, which yields the isometric realization of two-dimensional surfaces in $\R^3$. As a first application of this approach, we study the isometric immersion problem for two-dimensional Riemannian manifolds with strictly negative Gauss curvature. We prove that there exists a $C^{1,1}$ isometric immersion of the two-dimensional manifold in $\R^3$ satisfying our prescribed initial conditions. T
A vanishing viscosity method is formulated for two-dimensional transonic steady irrotational compressible fluid flows with adiabatic constant $γ\in [1,3)$. This formulation allows a family of invariant regions in the phase plane for the corresponding viscous problem, which implies an upper bound uniformly away from cavitation for the viscous approximate velocity fields. Mathematical entropy pairs are constructed through the Loewner-Morawetz relation by entropy generators governed by a generalized Tricomi equation of mixed elliptic-hyperbolic type, and the corresponding entropy dissipation measures are analyzed so that the viscous approximate solutions satisfy the compensated compactness framework. Then the method of compensated compactness is applied to show that a sequence of solutions to the artificial viscous problem, staying uniformly away from stagnation, converges to an entropy solution of the inviscid transonic flow problem.
In this paper, we present appendices employed in the paper "A geometric level-set formulation of a plasma-sheath interface" by the authors.