The phenomena of concentration and cavitation and the formation of δ-shocks and vacuum states in solutions to the Euler equations for isentropic fluids are identified and analyzed as the pressure vanishes. It is shown that, as the pressure vanishes, any two-shock Riemann solution to the Euler equations for isentropic fluids tends to a δ-shock solution to the Euler equations for pressureless fluids, and the intermediate density between the two shocks tends to a weighted δmeasure that forms the δ-shock. By contrast, any two-rarefaction-wave Riemann solution of the Euler equations for isentropic fluids is shown to tend to a two-contact-discontinuity solution to the Euler equations for pressureless fluids, whose intermediate state between the two contact discontinuities is a vacuum state, even when the initial data stays away from the vacuum. Some numerical results exhibiting the formation process of δ-shocks are also presented.

Abstract. We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic, O(1) critical threshold.We investigate a class of Euler-Poisson equations, ranging from one-dimensional problem with or without various forcing mechanisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time, the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds.A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock discontinuity except for the non-generic case of increasing initial profile, u ′ 0 ≥ 0. In contrast, we show that the corresponding one dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initial (ρ0, u0)-configuration satisfies u ′ 0 ≥ − √ 2kρ0 -see (2.11) below, allowing a finite, critical negative velocity gradient. As is typical for such nonlinear convection problems one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena.

The bile acid-activated receptors, nuclear farnesoid X receptor (FXR) and the membrane Takeda G-protein receptor 5 (TGR5), are known to improve glucose and insulin sensitivity in obese and diabetic mice. However, the metabolic roles of these two receptors and the underlying mechanisms are incompletely understood. Here, we studied the effects of the dual FXR and TGR5 agonist INT-767 on hepatic bile acid synthesis and intestinal secretion of glucagon-like peptide-1 (GLP-1) in wild-type, , and mice. INT-767 efficaciously stimulated intracellular Ca levels, cAMP activity, and GLP-1 secretion and improved glucose and lipid metabolism more than did the FXR-selective obeticholic acid and TGR5-selective INT-777 agonists. Interestingly, INT-767 reduced expression of the genes in the classic bile acid synthesis pathway but induced those in the alternative pathway, which is consistent with decreased taurocholic acid and increased tauromuricholic acids in bile. Furthermore, FXR activation induced expression of FXR target genes, including fibroblast growth factor 15, and unexpectedly and prohormone convertase 1/3 gene expression in the ileum. We identified an FXR-responsive element on the gene promoter. and mice exhibited reduced GLP-1 secretion, which was stimulated by INT-767 in the mice but not in the mice. Our findings uncovered a novel mechanism in which INT-767 activation of FXR induces gene expression and increases Ca levels and cAMP activity to stimulate GLP-1 secretion and improve hepatic glucose and lipid metabolism in high-fat diet-induced obese mice. Activation of both FXR and TGR5 may therefore represent an effective therapy for managing hepatic steatosis, obesity, and diabetes.

Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47 (1) (2009), . In this work, we show that higher order (k ≥ 4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all p k elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one-and two-dimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.

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