Smoothing properties and existence of solutions for the generalized Benjamin-Ono equation

Abstract We study the Cauchy problem for the generalized Benjamin-Ono equation ∂tu − D |D|2μ u = DV'(u), (∗) where D = d dx , U > 0 , and V ϵ E 1( R,R ) with V(0) = V′(0) = 0. Using commutator identities and estimates, we prove that the equation (∗) exhibits smoothing properties similar to those of the generalized Korteweg-de Vries equation (GKdV), which is the special case μ = 1 of (∗). If V satisfies suitable estimates at zero and at infinity, we prove that the Cauchy problem for the equation (∗) with initial data u(0) = u0 has a solution u ϵ L∞( R , l2)∪L2loc( R , Hμloc) if u0 ϵ L2, and has a solution u ϵ L∞( R , Hμ)∪L2loc( R , H2μloc)if u0 ϵ Hμ (finite energy solution). In addition, we prove that the usual Benjamin-Ono equation ( μ = 1 2 , V′(u) = u 2 ) has a solution u ϵ L∞( R , H1)∪L2loc( R , H 3 2 loc) for u0 ϵ H1. Those results are easily extended to related equations such as the intermediate long wave equation or the Smith equation

On the Cauchy Problem for the Zakharov System

Abstract We study the local Cauchy problem in time for the Zakharov system, (1.1) and (1.2), governing Langmuir turbulence, with initial data ( u (0), n (0), ∂ t n (0))∈ H k ⊕ H lscr; ⊕ H l−1 , in arbitrary space dimension ν . We define a natural notion of criticality according to which the critical values of ( k , l) are ( ν /2−3/2, ν /2−2). Using a method recently developed by Bourgain, we prove that the Zakharov system is locally well posed for a variety of values of ( k , l). The results cover the whole subcritical range for ν ⩾4. For ν ⩽3, they cover only part of it and the lowest admissible values are ( k , l)=(1/2, 0) for ν =2, 3 and ( k , l)=(0, −1/2) for ν =1. As a by product of the one dimensional result, we prove well-posedness of the Benney system, (1.14) and (1.15), governing the interaction of short and long waves for the same values of ( k , l)

Canagliflozin and renal outcomes in type 2 diabetes and nephropathy

BACKGROUND Type 2 diabetes mellitus is the leading cause of kidney failure worldwide, but few effective long-term treatments are available. In cardiovascular trials of inhibitors of sodium–glucose cotransporter 2 (SGLT2), exploratory results have suggested that such drugs may improve renal outcomes in patients with type 2 diabetes. METHODS In this double-blind, randomized trial, we assigned patients with type 2 diabetes and albuminuric chronic kidney disease to receive canagliflozin, an oral SGLT2 inhibitor, at a dose of 100 mg daily or placebo. All the patients had an estimated glomerular filtration rate (GFR) of 30 to <90 ml per minute per 1.73 m2 of body-surface area and albuminuria (ratio of albumin [mg] to creatinine [g], >300 to 5000) and were treated with renin–angiotensin system blockade. The primary outcome was a composite of end-stage kidney disease (dialysis, transplantation, or a sustained estimated GFR of <15 ml per minute per 1.73 m2), a doubling of the serum creatinine level, or death from renal or cardiovascular causes. Prespecified secondary outcomes were tested hierarchically. RESULTS The trial was stopped early after a planned interim analysis on the recommendation of the data and safety monitoring committee. At that time, 4401 patients had undergone randomization, with a median follow-up of 2.62 years. The relative risk of the primary outcome was 30% lower in the canagliflozin group than in the placebo group, with event rates of 43.2 and 61.2 per 1000 patient-years, respectively (hazard ratio, 0.70; 95% confidence interval [CI], 0.59 to 0.82; P=0.00001). The relative risk of the renal-specific composite of end-stage kidney disease, a doubling of the creatinine level, or death from renal causes was lower by 34% (hazard ratio, 0.66; 95% CI, 0.53 to 0.81; P<0.001), and the relative risk of end-stage kidney disease was lower by 32% (hazard ratio, 0.68; 95% CI, 0.54 to 0.86; P=0.002). The canagliflozin group also had a lower risk of cardiovascular death, myocardial infarction, or stroke (hazard ratio, 0.80; 95% CI, 0.67 to 0.95; P=0.01) and hospitalization for heart failure (hazard ratio, 0.61; 95% CI, 0.47 to 0.80; P<0.001). There were no significant differences in rates of amputation or fracture. CONCLUSIONS In patients with type 2 diabetes and kidney disease, the risk of kidney failure and cardiovascular events was lower in the canagliflozin group than in the placebo group at a median follow-up of 2.62 years