arXiv subjects

Lishan Liu

Publications and source records attributed to Lishan Liu.

3 recordsLinked to original sources

arXiv2023

Directional Differentiability of the Generalized Metric Projection in Hilbert spaces and Hilbertian Bochner spaces

Let $H$ be a real Hilbert space and $C$ a nonempty closed and convex subset of $H$. Let $P_C: H\rightarrow C$ denote the (standard) metric projection operator. In this paper, we study the Gâteaux directional differentiability of $P_C$ and investigate some of its properties. The Gâteaux directionally derivatives of $P_C$ are precisely given for the following cases of the considered subset $C$: 1. closed and convex subsets; 2. closed balls; 3. closed and convex cones (including proper closed subspaces). For special Hilbert spaces, we consider directional differentiability of $P_C$ for some Hilbert spaces with orthonormal bases and the real Hilbert space $L^2([-π,π])$ with the trigonometric orthonormal basis.

math.FA↗

arXiv2020

On the well-posedness and decay rates of strong solutions to a multi-dimensional non-conservative viscous compressible two-fluid system

The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a complicated parabolic-hyperbolic system, and on a refined time-weighted inequality.

math.AP↗

arXiv2018

Existence, uniqueness and optimal decay rates for the 3D compressible Hall-magnetohydrodynamic system

We are concerned with the study of the Cauchy problem to the 3D compressible Hall-magnetohydrodynamic system. We first establish the unique global solvability of strong solutions to the system when the initial data are close to a stable equilibrium state in critical Besov spaces. Furthermore, under a suitable additional condition involving only the low frequencies of the data and in $L^{2}$-critical regularity framework, we exhibit the optimal time decay rates for the constructed global solutions. The proof relies on an application of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined time-weighted energy functional.

math.AP↗