A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.2000 Mathematics Subject Classification. 05C45, 34L15, 35Pxx, 35R02.

This paper was prepared for presentation at the 47th Annual Fall Meeting of the Society of Petroleum Engineers held in San Antonio, Tex., Oct. 8–11, 1972. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by who the paper is presented. Publication elsewhere after publication in the JOURNAL paper is presented. Publication elsewhere after publication in the JOURNAL OF PETROLEUM TECHNOLOGY or the SOCIETY OF PETROLEUM ENGINEERS JOURNAL is usually granted upon request to the Editor of the appropriate journal provided agreement to give proper credit is made. provided agreement to give proper credit is made. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Introduction The oil industry has shown a great deal of interest in polymer solutions for use in recovery operations. A small amount of polymer in the injected water of flooding polymer in the injected water of flooding operations reduces the mobility of the driving phase. With the use of polymer solutions for phase. With the use of polymer solutions for recovery operations, the need has developed to understand and describe the mechanisms of non-Newtonian flow through porous media. Many investigators have studied the flow of fluids through porous media. Early investigators, such as Darcy and Kozeny, formulated equations for predicting pressure drop versus flow rate relationships for Newtonian fluids. In recent years, investigators have modified Darcy's equation to account for flow of purely viscous, non-Newtonian fluids. Some purely viscous, non-Newtonian fluids. Some workers have mentioned the effects of elasticity on flow through porous media. One rheological model has been reported where an empirical term is used to describe the effect of elasticity during flow through porous media. porous media. The present investigation shows how viscoelastic fluids behave while flowing through unconsolidated porous media. Details are given on the performance of various polymer solutions in a capillary flow device. polymer solutions in a capillary flow device. This instrument was used to determine if selected polymer solutions exhibited elastic properties. Rheological results are presented properties. Rheological results are presented for different polymer solutions. Flow results in porous media are given for these same polymer solutions. Rheological and flow results polymer solutions. Rheological and flow results are compared as a means of considering the effect of viscoelasticity on polymer solution flow through porous media. A modified form of Darcy's law is presented for use with viscoelastic liquids. SOLUTION CHARACTERIZATION Polymer solutions for this investigation were prepared as specified in Table 1. All polymers were mixed in distilled water. polymers were mixed in distilled water. Polymer was added slowly and the solution stirred Polymer was added slowly and the solution stirred gently by hand to avoid shear degradation. For the more concentrated solutions, the polymer addition period took up to one day. After polymer addition period took up to one day. After standing for a few days, solutions were filtered through a cotton cloth to remove any gelled material.

In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class.

In this paper, we draw on the ideas of [5] to extend the standard Serrin criterion [17] to an anisotropic version thereof. Because we work on weak solutions instead of strong ones, the functions involved have low regularity. Our method summarizes in a joint use of a uniqueness lemma in low regularity and the existence of stronger solutions. The uniqueness part uses duality in a way quite similar to the DiPerna-Lions theory, first developed in [7]. The existence part relies on L p energy estimates, whose proof may be found in [5], along with an approximation procedure.

In this paper, we extend our previous result from [24]. We prove that transport equations with rough coefficients do possess a uniqueness property, even in the presence of viscosity. Our method relies strongly on duality and bears a strong resemblance with the wellknown DiPerna-Lions theory first developed in [13]. This uniqueness result allows us to reprove the celebrated theorem of J. Serrin [28] in a novel way. As a byproduct of the techniques, we derive an L 1 bound for the vorticity in terms of a critical Lebesgue norm of the velocity field. We also show that the zero solution is unique for the 2D Euler equations on the torus under a mild integrability assumption. TODO : chercher diverseséquations classiques où les idées d'unicité s'appliquent