Weak in Space, Log in Time Improvement of the Lady{\v{z}}enskaja-Prodi-Serrin Criteria

In this article we present a Lady{\v{z}}enskaja-Prodi-Serrin Criteria for regularity of solutions for the Navier-Stokes equation in three dimensions which incorporates weak $L^p$ norms in the space variables and log improvement in the time variable.Comment: 14 pages, to appea

The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations

We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, i. e. \sup_{j\in\Z}\int_0^T\|\Delta_j(\na\times u)\|_\infty dt, where $\Delta_j$ is a frequency localization on $|\xi|\approx 2^j$.Comment: 12page

Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$

Conditional regularity of solutions of the three dimensional Navier-Stokes equations and implications for intermittency

Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on $L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$, and particularly on $D_{m} = \Omega_{m}^{\alpha_{m}}$, where $\alpha_{m} = 2m/(4m-3)$ for $m\geq 1$. The first result, more appropriate for the unforced case, can be stated simply : if there exists an $1\leq m < \infty$ for which the integral condition is satisfied ($Z_{m}=D_{m+1}/D_{m}$) $$\int_{0}^{t}\ln (\frac{1 + Z_{m}}{c_{4,m}}) d\tau \geq 0$$ then no singularity can occur on $[0, t]$. The constant $c_{4,m} \searrow 2$ for large $m$. Secondly, for the forced case, by imposing a critical \textit{lower} bound on $\int_{0}^{t}D_{m} d\tau$, no singularity can occur in $D_{m}(t)$ for \textit{large} initial data. Movement across this critical lower bound shows how solutions can behave intermittently, in analogy with a relaxation oscillator. Potential singularities that drive $\int_{0}^{t}D_{m} d\tau$ over this critical value can be ruled out whereas other types cannot.Comment: A frequency was missing in the definition of D_{m} in (I5) v3. 11 pages, 1 figur

Preface

Funder: Universität Konstanz (3156

Low Incidence of Chest Wall Pain with a Risk-Adapted Lung Stereotactic Body Radiation Therapy Approach Using Three or Five Fractions Based on Chest Wall Dosimetry

Purpose To examine the frequency and potential of dose-volume predictors for chest wall (CW) toxicity (pain and/or rib fracture) for patients receiving lung stereotactic body radiotherapy (SBRT) using treatment planning methods to minimize CW dose and a risk-adapted fractionation scheme. Methods: We reviewed data from 72 treatment plans, from 69 lung SBRT patients with at least one year of follow-up or CW toxicity, who were treated at our center between 2010 and 2013. Treatment plans were optimized to reduce CW dose and patients received a risk-adapted fractionation of 18 Gy×3 fractions (54 Gy total) if the CW V30 was less than 30 mL or 10–12 Gy×5 fractions (50–60 Gy total) otherwise. The association between CW toxicity and patient characteristics, treatment parameters and dose metrics, including biologically equivalent dose, were analyzed using logistic regression. Results: With a median follow-up of 20 months, 6 (8.3%) patients developed CW pain including three (4.2%) grade 1, two (2.8%) grade 2 and one (1.4%) grade 3. Five (6.9%) patients developed rib fractures, one of which was symptomatic. No significant associations between CW toxicity and patient and dosimetric variables were identified on univariate nor multivariate analysis. Conclusions: Optimization of treatment plans to reduce CW dose and a risk-adapted fractionation strategy of three or five fractions based on the CW V30 resulted in a low incidence of CW toxicity. Under these conditions, none of the patient characteristics or dose metrics we examined appeared to be predictive of CW pain

Generalised Gagliardo–Nirenberg inequalities using weak Lebesgue spaces and BMO

Using elementary arguments based on the Fourier transform we prove that for $1 \leq q n(1/2-1/p)$, if $f \in L^{q,\infty}(\R^n) \cap \dot{H}^s(\R^n)$ then $f \in L^p(\R^n)$ and there exists a constant $c_{p,q,s}$ such that $$\|f\|_{L^p} \leq c_{p,q,s} \|f\|_{L^{q,\infty}}^\theta \|f\|_{\dot H^s}^{1-\theta},$$ where $1/p = \theta/q + (1-\theta)(1/2-s/n)$. In particular, in $\R^2$ we obtain the generalised Ladyzhenskaya inequality $\|f\|_{L^4}\le c\|f\|_{L^{2,\infty}}^{1/2}\|f\|_{\dot H^1}^{1/2}$. We also show that for $s=n/2$ the norm in $\|f\|_{\dot H^{n/2}}$ can be replaced by the norm in BMO. As well as giving relatively simple proofs of these inequalities, this paper provides a brief primer of some basic concepts in harmonic analysis, including weak spaces, the Fourier transform, the Lebesgue Differentiation Theorem, and Calderon-Zygmund decompositions

Alliance Foundation Trial 09: A randomized, multicenter, phase 2 trial evaluating two sequences of pembrolizumab and standard platinum-based chemotherapy in patients with metastatic NSCLC

INTRODUCTION: The sequence of chemotherapy and pembrolizumab may affect antitumor immune response and efficacy of immunotherapy. METHODS: This multicenter, randomized, phase 2 trial was designed to evaluate the efficacy of two sequences of chemotherapy and pembrolizumab in patients with stage 4 NSCLC. Both arms were considered investigational, and the study used a pick a winner design. The primary end point was objective response rate by independent radiologic review after eight cycles (24 wk). Patients were randomized 1:1 to arm A (chemotherapy for four cycles followed by pembrolizumab for four cycles) or arm B (pembrolizumab for four cycles followed by chemotherapy for four cycles). Patients in both arms without disease progression after the initial eight cycles continued pembrolizumab until disease progression, unacceptable toxicity, or a maximum of 2 years. RESULTS: From March 2016 to July 2018, a total of 90 eligible patients were randomized (43 patients to arm A and 47 patients to arm B). The objective response rate at 24 weeks in arms A and B was 39.5 % (95 % confidence interval [CI]: 24.9%-54.1 %) and 40.4 % (95 % CI: 26.4%-54.5 %), respectively ( CONCLUSIONS: Additional evaluation of either sequence in a phase 3 trial is not warranted

On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations

We improve and extend some known regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations by means of the Fourier localization technique and Bony's para-product decomposition.Comment: 13page