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We consider the Cauchy problem to the barotropic compressible Navier-Stokes equations. We obtain optimal local well-posedness in the sense of Hadamard in the critical Besov space $\mathbb{X}_p=\dot{B}_{p,1}^{\frac{d}{p}}\times \dot{B}_{p,1}^{-1+\frac{d}{p}}$ for $1\leq p<2d$ with $d\geq2$. The main new result is the continuity of the solution maps from $\mathbb{X}_p$ to $C([0,T]: \mathbb{X}_p)$, which was not proved in previous works \cite{D2001, D2005, D2014}. To prove our results, we combine the method of frequency envelope (see \cite{Tao04}) but in the transport-parabolic setting and the Lagrangian approach for the compressible Navier-Stokes equations (see \cite{D2014}). As a by-product, the Lagrangian transform $(a,u)\to (\bar a, \bar u)=(a\circ X, u\circ X)$ used in \cite{D2014} is a continuous bijection and hence bridges the two methods using Eulerian and Lagrangian coordinates.

Multimodal sentiment analysis (MSA) is an important way of observing mental activities with the help of data captured from multiple modalities. However, due to the recording or transmission error, some modalities may include incomplete data. Most existing works that address missing modalities usually assume a particular modality is completely missing and seldom consider a mixture of missing across multiple modalities. In this paper, we propose a simple yet effective meta-sampling approach for multimodal sentiment analysis with missing modalities, namely Missing Modality-based Meta Sampling (M3S). To be specific, M3S formulates a missing modality sampling strategy into the modal agnostic meta-learning (MAML) framework. M3S can be treated as an efficient add-on training component on existing models and significantly improve their performances on multimodal data with a mixture of missing modalities. We conduct experiments on IEMOCAP, SIMS and CMU-MOSI datasets, and superior performance is achieved compared with recent state-of-the-art methods.

Click-Through Rate (CTR) prediction has been an indispensable component for many industrial applications, such as recommendation systems and online advertising. CTR prediction systems are usually based on multi-field categorical features, i.e., every feature is categorical and belongs to one and only one field. Modeling feature conjunctions is crucial for CTR prediction accuracy. However, it requires a massive number of parameters to explicitly model all feature conjunctions, which is not scalable for real-world production systems. In this paper, we describe a novel Field-Leveraged Embedding Network (FLEN) which has been deployed in the commercial recommender system in Meitu and serves the main traffic. FLEN devises a field-wise bi-interaction pooling technique. By suitably exploiting field information, the field-wise bi-interaction pooling captures both inter-field and intra-field feature conjunctions with a small number of model parameters and an acceptable time complexity for industrial applications. We show that a variety of state-of-the-art CTR models can be expressed under this technique. Furthermore, we develop Dicefactor: a dropout technique to prevent independent latent features from co-adapting. Extensive experiments, including offline evaluations and online A/B testing on real production systems, demonstrate the effectiveness and efficiency of FLEN against the state-of-the-arts. Notably, FLEN has obtained 5.19% improvement on CTR with 1/6 of memory usage and computation time, compared to last version (i.e. NFM)., Comment: KDD'20 DLP workshop, 9 pages

In this paper, we construct a class of global large solution to the three-dimensional Navier-Stokes equations with the Coriolis force in critical Fourier-Besov space $\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)$. In fact, our choice of special initial data $u_0$ can be arbitrarily large in $\dot{FB}^{s}_{p,r}(\mathbb{R}^3)$ for any $s\in\mathbb{R}$ and $1\leq p,r\leq \infty$.

In this paper, we construct a class global large solution to the two-dimensional MHD equations with damp terms in the nonhomogeneous Sobolev framework.

Let $L$ be a Neumann operator of the form $L=-\Delta_{N}$ acting on $L^2(\mathbb R^n)$. Let ${BMO}_{\Delta_{N}}(\mathbb R^n)$ denote the BMO space on $\mathbb R^n$ associated to the Neumann operator $\L$. In this article we will show that a function $f\in { BMO}_{\Delta_{N}}(\mathbb R^n)$ is the trace of the solution of $${\mathbb L}u=u_{t}+L u=0, u(x,0)= f(x),$$ where $u$ satisfies a Carleson-type condition \begin{eqnarray*} \sup_{x_B, r_B} r_B^{-n}\int_0^{r_B^2}\int_{B(x_B, r_B)} |\nabla u(x,t)|^2 {dx dt } \leq C <\infty, \end{eqnarray*} for some constant $C>0$. Conversely, this Carleson condition characterizes all the ${\mathbb L}$-carolic functions whose traces belong to the space ${BMO}_{\Delta_{N}}(\mathbb R^n)$. This result extends the analogous characterization founded by E. Fabes and U. Neri in ({Duke Math. J.} {42} (1975), 725-734) for the classical BMO space of John and Nirenberg. Furthermore, based on the characterization of ${BMO}_{\Delta_{N}}(\mathbb R^n)$ space mentioned above, we prove global well-posedness for parabolic equations of Navier-Stokes type with the Neumann boundary condition under smallness condition on intial data $u_{0}\in {{ BMO}_{\Delta_{N}}^{-1}(\mathbb R^n)}$, which is motivated by the work of P. Auscher and D. Frey ({J. Inst. Math. Jussieu} {16(5)} (2017), 947-985)., Comment: 14 pages

Let L be a Schr\"odinger operator of the form L=-\Delta+V acting on L^2(\mathbb R^n) where the nonnegative potential V belongs to the reverse H\"older class B_q for some q>= n. Let BMO denote the BMO space associated to the Schr\"odinger operator L. In this article we will show that a function f in BMO_L is the trace of the solution of u_t+L u=0, u(x,0)= f(x), where u satisfies a Carleson-type condition. Conversely, this Carleson condition characterizes all the L-carolic functions whose traces belong to the space BMO_L. This result extends the analogous characterization founded by Fabes and Neri for the classical BMO space of John and Nirenberg., Comment: 22 pages