BACKGROUND: Despite known risks of using chronic opioid therapy (COT) for pain, the risks of discontinuation of COT are largely uncharacterized. OBJECTIVE: To evaluate mortality, prescription opioid use, and primary care utilization of patients discontinued from COT, compared with patients maintained on opioids. DESIGN: Retrospective cohort study of patients with chronic pain enrolled in an opioid registry as of May 2010. PARTICIPANTS: Patients with chronic pain enrolled in the opioid registry of a primary care clinic at an urban safety-net hospital in Seattle, WA. MAIN OUTCOMES AND MEASURES: Discontinuation from the opioid registry was the exposure of interest. Pre-specified main outcomes included mortality, prescription and primary care utilization data, and reasons for discontinuation. Data was collected through March 2015. KEY RESULTS: The study cohort comprised 572 patients with a mean age of 54.9 ± 10.1 years. COT was discontinued in 344 patients (60.1%); 254 (73.8%) discontinued patients subsequently filled at least one opioid prescription in Washington State, and 187 (54.4%) continued to visit the clinic. During the study period, 119 (20.8%) registry patients died, and 21 (3.7%) died of definite or possible overdose: 17 (4.9%) discontinued patients died of overdose, whereas 4 (1.75%) retained patients died of overdose. Most patients had at least one provider-initiated reason for COT discontinuation. Discontinuation of COT was associated with a hazard ratio for death of 1.35 (95% CI, 0.92 to 1.98, p = 0.122) and for overdose death of 2.94 (1.01-8.61, p = 0.049), after adjusting for age and race. CONCLUSIONS: In this cohort of patients prescribed COT for chronic pain, mortality was high. Discontinuation of COT did not reduce risk of death and was associated with increased risk of overdose death. Improved clinical strategies, including multimodal pain management and treatment of opioid use disorder, may be needed for this highrisk group.

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [31]. Consider a passive scalar in a shear flow, whose evolution is described by the advection-diffusion equationwith periodic, Dirichlet, or Neumann conditions in y. We demonstrate that decay is enhanced on the timescale T ∼ ν −(N+1)/(N+3) , where N −1 is the maximal order of vanishing of the derivative b ′ (y) of the shear profile and N = 0 for monotone shear flows. In the periodic setting, we recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are new in the presence of boundaries.

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The purpose of this note is to study the weak solutions to the inviscid quasigeostrophic system for initial data belonging to Lebesgue spaces. We give a global existence result as well as detail the connections between several different notions of weak solutions. In addition, we give a condition under which the energy of the system is conserved. 1 2 is the Dirichlet-to-Neumann operator for R 3 + , we set for each time t ≥ 0where R 1 , R 2 are the Riesz transforms in R 2 . Then (QG) reduces to the well-studied inviscid surface quasi-geostrophic equation, which can be written asFor the sake of consistency and to keep in mind the connection to the 3D model, we shall always treat ∇, ∇ ⊥ , and R ⊥ as vectors with three components and zero first component. SQG has received considerable attention due to its similarities with the important systems of fluid mechanics (see Constantin, Majda, and Tabak [7], Garner, Held, Pierrehumber, and Swanson [14], among others). Weak solutions were constructed in L 2 by Resnick [23]. Marchand [18] first gave a proof of the existence of global weak solutions when the initial data is not in L 2 but rather L p for any p > 4 3 . When critical dissipation is added to the transport equation for ∂ ν Ψ in (QG), global regularity was established in [21] using the De Giorgi technique in combination with a bootstrapping argument and an appropriate Beale-Kato-Majda criterion. Surface quasi-geostrophic flows on bounded domains have been considered by Constantin and Ignatova [9], [8], Constantin and Nguyen [10], [11] and Nguyen [19] using the spectral Riesz transform. Global existence of weak solutions for inviscid SQG is shown by Constantin and Nguyen [10] and for a generalized SQG model by Nguyen [19]. In [20], an appropriate boundary condition is derived and global weak solutions to the 3D model posed on a cylindrical domain are constructed.1.1. The Reformulated System. A crucial tool in our analysis will be a reformulation of (QG). We draw inspiration from Puel and Vasseur [22], who used a reformulation to obtain their global existence result. The physical system as written is analogous to the vorticity form of the Euler equations with an additional boundary condition. However, one may consider the following reformulation, in which curl(Q) acts as a Lagrange multiplier similar to the gradient of the pressure in the Euler equations:Formally, taking the divergence of (rQG) gives (QG) L , and taking the trace gives (QG) ν . To obtain (rQG) from (QG), one must invert the divergence operator coupled with a Neumann boundary condition. While providing a link between the two formulations will be an important part of our analysis (see Theorem 1.3), let us proceed from the perspective of (rQG) for the time being. Following Puel and Vasseur [22], we define the notion of weak solutions to (rQG). Definition 1.1 (Weak Solutions to (rQG)). Let T, R be fixed, φ ∈ C ∞ (R 4 ) compactly supported in (−T, T ) × (−R, R) 3 , and F be such that ∆F = f L , ∂ ν F = f ν . A weak solution Ψ to (rQG) with forcing f ν , f L...