5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

Valuable data on quarkonia (the bound states of a heavy quark $Q=c,b$ and the corresponding antiquark) have recently been provided by a variety of sources, mainly ${e}^{+}{e}^{\ensuremath{-}}$ collisions, but also hadronic interactions. This permits a thorough updating of the experimental and theoretical status of electromagnetic and strong transitions in quarkonia. The $Q\overline{Q}$ transitions to other $Q\overline{Q}$ states are discussed, with some reference to processes involving $Q\overline{Q}$ annihilation.

Quarkonia and their transitions

Hocherman and Rosenau conjectured that long-wave unstable Cahn-Hilliard type interface models develop nite-time singularities when the nonlinearity in the destabilizing term grows faster at large amplitudes than the nonlinearity in the stabilizing term (Phys. D 67:113-125, 1993). We consider this conjecture for a class of equations, often used to model thin lms in a lubrication context, by showing that if the solutions are uniformly bounded above or below (e.g. are nonnegative) then the destabilizing term can be stronger than previously conjectured yet the solution still remains globally bounded. For example, they conjecture that for the long-wave unstable equation ht = (h n hxxx)x (h m hx)x; m>n leads to blow-up. Using a conservation of volume constraint for the case m> n> 0, we conjecture a dierent critical exponent for possible singularities of nonnegative solutions. We prove that nonlinearities with exponents below the conjectured critical exponent yield globally bounded solutions. Specically, for the above equation, solutions are bounded if m<n + 2. The bound is proved using energy methods and is then used to prove the existence of non-negative weak solutions in the sense of distributions. We present preliminary numerical evidence suggesting that m n +2 can allow blow-up.

/pdf/long-wave-instabilities-and-saturation-in-thin-film-1k9ny0qeac.pdf

Long-wave instabilities and saturation in thin film equations

This chapter discusses global attractors in partial differential equations (PDE). Compact global attractors are robust objects with respect to perturbations. In general, the invariance and attractivity of the global attractors, combined with some additional hypotheses, imply interesting robustness and regularity properties. Often also, the flow restricted to the global attractor shows finite dimensional behavior. If the semigroups are defined on a finite or infinite dimensional vector space, for example, then considerable care must be taken to discuss the behavior of orbits at infinity. If each of the semigroups has a compact global attractor, the topological equivalence of the flows restricted to the global attractors can be considered. This is the strongest type of comparison of flows that can be expected in that it uses the very detailed properties of the flows. In the chapter, weaker concepts of comparison—such as estimates of the Hausdorff distance between the global attractors—are considered.

Chapter 17 - Global Attractors in Partial Differential Equations

This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and bi-Laplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to present some new ones. Some of the names associated with these inequalities are Rayleigh, Faber-Krahn, Szego-Weinberger, Payne-Polya-Weinberger, Sperner, Hile-Protter, and H.C. Yang. Occasionally, we will also comment on extensions of some of our inequalities to bounded domains in other spaces, specifically, S n or H n . Introduction The Eigenvalue Problems The first eigenvalue problem we shall introduce is that of the fixed membrane, or Dirichlet Laplacian . We consider the eigenvalues and eigenfunctions of –Δ on a bounded domain (=connected open set) Ω in Euclidean space R n , i.e., the problem It is well-known that this problem has a real and purely discrete spectrum where Here each eigenvalue is repeated according to its multiplicity. An associated orthonormal basis of real eigenfunctions will be denoted u 1 , u 2 , u 3 , …. In fact, throughout this paper we will assume that all functions we consider are real. This entails no loss of generality in the present context. The next problem we introduce is that of the free membrane, or Neumann Laplacian .

Isoperimetric and universal inequalities for eigenvalues

New bounds are given for the L2-norm of the solution of the Kuramoto-Sivashinsky equation

$$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$

, for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is

$$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$

.

/pdf/a-global-attracting-set-for-the-kuramoto-sivashinsky-2xrh539s6o.pdf

A global attracting set for the Kuramoto-Sivashinsky equation

In this article, we prove and exploit a trace identity for the spectra of Schrodinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang. Introduction In this article, we prove and exploit an identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (p−A(x))2 + V (x), (1) where p = 1i ∇ is the usual momentum operator in convenient units and A(x) is the magnetic vector potential. We recover and extend several known inequalities involving sums, differences, and ratios of eigenvalues. Let λj , j = 1, . . . , denote the ordered eigenvalues of the Dirichlet Laplacian on a bounded d-dimensional domain with zero Dirichlet boundary conditions, and recall that Hile and Protter [HiPr80] proved that: d 4 ≤ 1 n n ∑ j=1 λj λn+1 − λj , (2) thereby extending an earlier inequality of Payne, Polya, and Weinberger [PaPoWe56]. In the last few years it has become clear that these and many similar relationships can be realized as special cases of abstract variational bounds involving the interplay among commutators of −∇2, a Cartesian coordinate xj , and the corresponding derivative ∂/∂xj. For this analysis and various extensions of (2) see [Ha88], [Ho90], [Ha93], [HaMi95]. While inequalities of this type have been fairly sharp for low-lying eigenvalues, they have been mostly disappointing for higher eigenvalues. Yang [Ya91], however, Received by the editors September 28, 1995. 1991 Mathematics Subject Classification. Primary 35J10, 35J25, 58G25.

https://www.ams.org/tran/1997-349-05/S0002-9947-97-01846-1/S0002-9947-97-01846-1.pdf

On trace identities and universal eigenvalue estimates for some partial differential operators

We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space, which extend those known previously for Laplacians and Schrodinger operators, freeing them from restrictive assumptions on the nature of the spectrum and allowing operators of much more general form. In particular, we allow for the presence of continuous spectrum, which is a necessary underpinning for new proofs of Lieb-Thirring inequalities. We both sharpen and extend universal bounds on spectral gaps and moments of eigenvalues {lambda(k)} of familiar types, and in addition we produce novel kinds of inequalities that are new even in the model cases. These include a family of differential inequalities for generalized Riesz means and a theorem stating that arithmetic means of {lambda(p)(k)}(k=1)(n) with p 8.

Universal Bounds and Semiclassical Estimates for Eigenvalues of Abstract Schrödinger Operators

We consider the stability problem for the localized solutions of classical nonlinear spinor fields in space dimensions N=1 and N=3 within the framework of the Shatah-Strauss formalism. We show that the well-known relations existing between the different stability criteria for scalar field equations are no longer valid for spinor fields. We discuss the application of the Shatah-Strauss formalism to several models: e.g., the Gross-Neveu, Thirring, and the Soler models.

Stability of nonlinear spinor fields with application to the Gross-Neveu model.

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kroger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schrodinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.

https://hal.archives-ouvertes.fr/hal-01174814/document

Joachim Stubbe

Papers

A global attracting set for the Kuramoto-Sivashinsky equation

On trace identities and universal eigenvalue estimates for some partial differential operators

Universal Bounds and Semiclassical Estimates for Eigenvalues of Abstract Schrödinger Operators

Stability of nonlinear spinor fields with application to the Gross-Neveu model.

On sums of eigenvalues of elliptic operators on manifolds