A theory of L1L^1-dissipative solvers for scalar conservation laws with discontinuous flux

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes

Low-frequency and rare-coding variation contributes to multiple sclerosis risk

Multiple sclerosis is a complex neurological disease, with ∼20% of risk heritability attributable to common genetic variants, including >230 identified by genome-wide association studies. Multiple strands of evidence suggest that much of the remaining heritability is also due to additive effects of common variants rather than epistasis between these variants or mutations exclusive to individual families. Here, we show in 68,379 cases and controls that up to 5% of this heritability is explained by low-frequency variation in gene coding sequence. We identify four novel genes driving MS risk independently of common-variant signals, highlighting key pathogenic roles for regulatory T cell homeostasis and regulation, IFNγ biology, and NFκB signaling. As low-frequency variants do not show substantial linkage disequilibrium with other variants, and as coding variants are more interpretable and experimentally tractable than non-coding variation, our discoveries constitute a rich resource for dissecting the pathobiology of MS

Low-frequency and rare-coding variation contributes to multiple sclerosis risk

Multiple sclerosis is a complex neurological disease, with ∼20% of risk heritability attributable to common genetic variants, including >230 identified by genome-wide association studies. Multiple strands of evidence suggest that much of the remaining heritability is also due to additive effects of common variants rather than epistasis between these variants or mutations exclusive to individual families. Here, we show in 68,379 cases and controls that up to 5% of this heritability is explained by low-frequency variation in gene coding sequence. We identify four novel genes driving MS risk independently of common-variant signals, highlighting key pathogenic roles for regulatory T cell homeostasis and regulation, IFNγ biology, and NFκB signaling. As low-frequency variants do not show substantial linkage disequilibrium with other variants, and as coding variants are more interpretable and experimentally tractable than non-coding variation, our discoveries constitute a rich resource for dissecting the pathobiology of MS

Erratum: Low-Frequency and Rare-Coding Variation Contributes to Multiple Sclerosis Risk (Cell (2019) 178(1) (262), (S0092867419306798), (10.1016/j.cell.2019.06.016))

(Cell 175, 1679–1687.e1–e7; November 29, 2018) It has come to our attention that in preparing the final version of this article, the authors inadvertently misspelled the last name of author Charlotte E. Teunissen as “Charlotte E. Theunissen.” This error has been corrected in the article online. In the Editorial Note (Cell 178, 262, June 27, 2019), the editors refer to the original version of the published manuscript. That version contained a misspelled name, and as that has now been corrected, we are updating the Editorial Note as well