An application of maximal dissipative sets in control theory

AbstractA model of a distributed-boundary control system is considered. Assume the uncontrolled system possesses an exponential asymptotically stable zero solution. We then construct suboptimal feedback controls for the distributed and boundary control problems via the direct method of Liapunov. Furthermore, existence-uniqueness of the synthesized control systems is proven by applying the theory of nonlinear semigroups and maximal dissipative sets. Applications to diffusion equations are given

Constitutive Relations for Rapid Granular Flow of Smooth Spheres: Generalized Rational Approximation to the Sum of the Chapman–Enskog Expansion

AbstractN. Sela and I. Goldhirsch (1998, J. Fluid Mech.361, 41–74) have used the Chapman–Enskog expansion to derive constitutive relations for the pressure deviator P, heat flux q, and rate of energy loss Γ for rapid flows of smooth inelastic spheres. Unfortunately as in the classical Chapman–Enskog expansion for elastic spheres any truncation of the expansion beyond Navier–Stokes order (n=1) will possess unphysical instabilities. This paper uses the method of approximate summation of the Chapman–Enskog expansion presented by the author (in press, Arch. Rational Mech. Anal.) to eliminate these instabilities

Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding

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

From the Nash--Kuiper Theorem to the Euler Equations of Fluid Dynamics

Direct linkages from the isometric embeddings of Riemannian manifolds to the compressible fluid dynamics are established. More precisely, let $(M,g)$ be a surface isometrically embedded in $\mathbb{R}^3$; by defining the density $\rho$, velocity $v$ and pressure $p$ in terms of the second fundamental form of the embedding, we get a solution for the steady compressible Euler equations of fluid dynamics. We also introduce a renormalization process to obtain solutions for Euler equations from non-$C^2$ isometric embeddings of the flat torus. Extensions to multi-dimensions are discussed.Comment: 2 figures deleted due to the size limit of arXiv; the figures are taken from the webpage of Dr. Vincent Borrelli: math.univ-lyon1.fr/homes-www/borrelli

Isometric embedding via strongly symmetric positive systems

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.Comment: 39 page

A Fluid Dynamic Formulation of the Isometric Embedding Problem in Differential Geometry

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.Comment: arXiv admin note: substantial text overlap with arXiv:0805.243