Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities

In this paper, we prove the local well-posedness in critical Besov spaces for the compressible Navier-Stokes equations with density dependent viscosities under the assumption that the initial density is bounded away from zero.Comment: 27page

A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation

We show a new Bernstein's inequality which generalizes the results of Cannone-Planchon, Danchin and Lemari\'{e}-Rieusset. As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data.Comment: 18page

Stress concentration for nonlinear insulated conductivity problem with adjacent inclusions

A high-contrast two-phase nonlinear composite material with adjacent inclusions of $m$-convex shapes is considered for $m>2$. The mathematical formulation consists of the insulated conductivity problem with $p$-Laplace operator in $\mathbb{R}^{d}$ for $p>1$ and $d\geq2$. The stress, which is the gradient of the solution, always blows up with respect to the distance $\varepsilon$ between two inclusions as $\varepsilon$ goes to zero. We first establish the pointwise upper bound on the gradient possessing the singularity of order $\varepsilon^{-\beta}$ with $\beta=(1-\alpha)/m$ for some $\alpha\geq0$, where $\alpha=0$ if $d=2$ and $\alpha>0$ if $d\geq3$. In particular, we give a quantitative description for the range of horizontal length of the narrow channel in the process of establishing the gradient estimates, which provides a clear understanding for the applied techniques and methods. For $d\geq2$, we further construct a supersolution to sharpen the upper bound with any $\beta>(d+m-2)/(m(p-1))$ when $p>d+m-1$. Finally, a subsolution is also constructed to show the almost optimality of the blow-up rate $\varepsilon^{-1/\max\{p-1,m\}}$ in the presence of curvilinear squares. This fact reveals a novel dichotomy phenomena that the singularity of the gradient is uniquely determined by one of the convexity parameter $m$ and the nonlinear exponent $p$ except for the critical case of $p=m+1$ in two dimensions

On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces

We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space (\dot{B}^{\f 3p}_{p,1}+\bar{\rho},\,\dot{B}^{\f 3p-1}_{p,1}) for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\bar{\rho}$ is a positive constant. Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space (\dot{B}^{\f 3p}_{p,1}+\bar{\rho},\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1}) for $p>3$. These results shows that the compressible Navier-Stokes equations are ill-posed in the smaller critical spaces compared with the incompressible Navier-Stokes equations.Comment: 25page

Well-posedness for the viscous shallow water equations in critical spaces

In this paper, we prove the existence and uniqueness of the solutions for the 2D viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.Comment: 32 page

Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data

In this paper, we prove the global well-posedness for the 3D rotating Navier-Stokes equations in the critical functional framework. Especially, this result allows to construct global solutions for a class of highly oscillating initial data.Comment: 20page

Global well-posedness for the micropolar fluid system in the critical Besov spaces

We prove the global well-posedness for the 3-D micropolar fluid system in the critical Besov spaces by making a suitable transformation to the solutions and using the Fourier localization method, especially combined with a new $L^p$ estimate for the Green matrix to the linear system of the transformed equation. This result allows to construct global solutions for a class of highly oscillating initial data of Cannone's type. Meanwhile, we analyze the long behavior of the solutions and get some decay estimates.Comment: 23page

Gut microbiota alterations are associated with functional outcomes in patients of acute ischemic stroke with non-alcoholic fatty liver disease

IntroductionPatients with acute ischemic stroke (AIS) with non-alcoholic fatty liver disease (NAFLD) frequently have poor prognosis. Many evidences suggested that the changes in gut microbiota may play an important role in the occurrence and development of AIS patients with NAFLD. The purpose of this study was to explore microbial characteristics in patients of AIS with NAFLD, and the correlation between gut microbiota and functional outcomes.MethodsThe patients of AIS were recruited and divided into NAFLD group and non-NAFLD group. The stool samples and clinical information were collected. 16 s rRNA sequencing was used to analyze the characteristics of gut microbiota. The patients of AIS with NAFLD were followed-up to evaluate the functional outcomes of disease. The adverse outcomes were determined by modified Rankin scale (mRS) scores at 3 months after stroke. The diagnostic performance of microbial marker in predicting adverse outcomes was assessed by recipient operating characteristic (ROC) curves.ResultsOur results showed that the composition of gut microbiota between non-NAFLD group and NAFLD group were different. The characteristic bacteria in the patients of AIS with NAFLD was that the relative abundance of Dorea, Dialister, Intestinibacter and Flavonifractor were decreased, while the relative abundance of Enorma was increased. Moreover, the characteristic microbiota was correlated with many clinical parameters, such as mRS scores, mean arterial pressure and fasting blood glucose level. In addition, ROC models based on the characteristic microbiota or the combination of characteristic microbiota with independent risk factors could distinguish functional dependence patients and functional independence patients in AIS with NAFLD (area under curve is 0.765 and 0.882 respectively).ConclusionThese findings revealed the microbial characteristics in patients of AIS with NAFLD, and further demonstrated the predictive capability of characteristic microbiota for adverse outcomes in patients of AIS with NAFLD