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In this paper we aim to construct a very weak solution to the steady two-dimensional Navier-Stokes equations which is affected by an external force induced by a point vortex on the unit disk. Such a solution is also the form of Prandtl-Batchelor type, i.e. the vorticity in the limit of vanishing viscosity is constant in an inner region separated from the boundary layer. Multi-scale asymptotic analysis which will capture well the singular behavior of the solution is used to construct higher order approximate solutions of the Navier-Stokes equations firstly, and then stability analysis is performed for the error system, finally the existence of Prandtl-Batchelor flows with a point vortex on a disk with the wall velocity slightly different from the rigid-rotation is proved. To overcome the singularity at the origin from the point vortex and strong singular behaviors near the boundary from viscous flow, we introduce a coordinate transformation which converts a bounded singularity problem to an unbounded one, and use a new decomposition for the error and perform basic energy estimate and positivity estimate for each part of the decomposition separately. The equation in the vorticity form shall be used to deal with $L^\infty$ estimate of the vorticity error., Comment: 49 pages

In this paper, we establish vanishing viscosity limit of the 2D Navier-Stokes equations in a horizontally periodic strip. On the vertical direction, the horizontal component of the velocity is subjected to two different types of boundary conditions: at the lower boundary, we give the degenerate zero boundary condition, while at the upper boundary, a small smooth perturbation of non-zero constant is prescribed. Due to this different boundary condition setting, the boundary layer effects are different near the lower and upper boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Couette flow $(Ay,0)$ for some suitable constant $A$, which is determined by using the principle of the Prandtl-Batchelor theory., Comment: Rewrite Lemma 2.3 and revise some stability estimates

This paper is concerned with blow-up solutions of the 4-dimensional energy critical heat equation $u_t=\Delta u + u^3$. Our main result is to show that the existence of type II blowup solutions, and $\|u(\cdot,t)\|_\infty\sim(T-t)^{-k}|ln(T-t)|^{\frac{2k}{2k-1}}, (k=1,2,3\cdots)$.The inner-outer gluing method has been employed., Comment: I have seen a paper with the same result published before

Simultaneous segmentation and detection of liver tumors (hemangioma and hepatocellular carcinoma (HCC)) by using multi-modality non-contrast magnetic resonance imaging (NCMRI) are crucial for the clinical diagnosis. However, it is still a challenging task due to: (1) the HCC information on NCMRI is invisible or insufficient makes extraction of liver tumors feature difficult; (2) diverse imaging characteristics in multi-modality NCMRI causes feature fusion and selection difficult; (3) no specific information between hemangioma and HCC on NCMRI cause liver tumors detection difficult. In this study, we propose a united adversarial learning framework (UAL) for simultaneous liver tumors segmentation and detection using multi-modality NCMRI. The UAL first utilizes a multi-view aware encoder to extract multi-modality NCMRI information for liver tumor segmentation and detection. In this encoder, a novel edge dissimilarity feature pyramid module is designed to facilitate the complementary multi-modality feature extraction. Second, the newly designed fusion and selection channel is used to fuse the multi-modality feature and make the decision of the feature selection. Then, the proposed mechanism of coordinate sharing with padding integrates the multi-task of segmentation and detection so that it enables multi-task to perform united adversarial learning in one discriminator. Lastly, an innovative multi-phase radiomics guided discriminator exploits the clear and specific tumor information to improve the multi-task performance via the adversarial learning strategy. The UAL is validated in corresponding multi-modality NCMRI (i.e. T1FS pre-contrast MRI, T2FS MRI, and DWI) and three phases contrast-enhanced MRI of 255 clinical subjects. The experiments show that UAL has great potential in the clinical diagnosis of liver tumors.

We are concerned with blow-up solutions of the 5-dimensional energy critical heat equation $u_t=\Delta u + | u |^{\frac{4}{3}}u$. Our main result is to show that the existence of type 2 solutions blows up at 2 points with 2 different blow-up rates. The inner-outer gluing method has been employed.