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Shumao Wang

Publications and source records attributed to Shumao Wang.

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Blow-up of the 3-D compressible Navier-Stokes equations for monatomic gases

In this paper, we prove the blow-up of the $3$-D isentropic compressible Navier-Stokes equations for the adiabatic exponent $γ=5/3$, which corresponds to the law of monatomic gases. This is the degenerate case in the sense of [Merle, Raphaël, Rodnianski and Szeftel, Ann. of Math. (2), 196 (2022), 567-778; Ann. of Math. (2), 196 (2022), 779-889]. Motivated by these breakthrough works, we first establish the existence of a sequence of smooth, self-similar imploding solutions to the compressible Euler equations for $γ=5/3$. Subsequently, we utilize these self-similar profiles to construct smooth, asymptotically self-similar blow-up solutions to the compressible Navier-Stokes equations for monatomic gases.

math.AP

*: Improving the 3D detector by introducing Voxel2Pillar feature encoding and extracting multi-scale features

The multi-line LiDAR is widely used in autonomous vehicles, so point cloud-based 3D detectors are essential for autonomous driving. Extracting rich multi-scale features is crucial for point cloud-based 3D detectors in autonomous driving due to significant differences in the size of different types of objects. However, because of the real-time requirements, large-size convolution kernels are rarely used to extract large-scale features in the backbone. Current 3D detectors commonly use feature pyramid networks to obtain large-scale features; however, some objects containing fewer point clouds are further lost during down-sampling, resulting in degraded performance. Since pillar-based schemes require much less computation than voxel-based schemes, they are more suitable for constructing real-time 3D detectors. Hence, we propose the *, a pillar-based scheme. We redesigned the feature encoding, the backbone, and the neck of the 3D detector. We propose the Voxel2Pillar feature encoding, which uses a sparse convolution constructor to construct pillars with richer point cloud features, especially height features. The Voxel2Pillar adds more learnable parameters to the feature encoding, enabling the initial pillars to have higher performance ability. We extract multi-scale and large-scale features in the proposed fully sparse backbone, which does not utilize large-size convolutional kernels; the backbone consists of the proposed multi-scale feature extraction module. The neck consists of the proposed sparse ConvNeXt, whose simple structure significantly improves the performance. We validate the effectiveness of the proposed * on the Waymo Open Dataset, and the object detection accuracy for vehicles, pedestrians, and cyclists is improved. We also verify the effectiveness of each proposed module in detail through ablation studies.

cs.CV

A slow blow up solution for the four dimensional energy critical semi linear heat equation

We consider the energy critical four dimensional semi-linear heat equation \[ \partial_{t}v-Δv-v^{3}=0, \quad(t,x)\in \mathbb{R}\times \mathbb{R}^4. \] Formal computation of Filippas et al. (R. Soc. Lond. Proc. 2000) conjectures the existence of a sequence of type II blow-up solutions with various blow-up rates \[ \|v(t)\|_{L^\infty(\mathbb{R}^4)}\approx \frac{|\log(T-t)|^{\frac{2L}{2L-1}}}{(T-t)^L} ,\quad L=1,2,\cdots.\] Schweyer (J. Funct. Anal. 2012) rigorously constructs a type II blow-up solution for the case $L=1$. In this paper, we show the existence of type II blow-up solution for $L=2$. The method here could be generalized to deal with all the cases $L\geq 2$.

math.AP