- Journal Home
- Volume 37 - 2024
- Volume 36 - 2023
- Volume 35 - 2022
- Volume 34 - 2021
- Volume 33 - 2020
- Volume 32 - 2019
- Volume 31 - 2018
- Volume 30 - 2017
- Volume 29 - 2016
- Volume 28 - 2015
- Volume 27 - 2014
- Volume 26 - 2013
- Volume 25 - 2012
- Volume 24 - 2011
- Volume 23 - 2010
- Volume 22 - 2009
- Volume 21 - 2008
- Volume 20 - 2007
- Volume 19 - 2006
- Volume 18 - 2005
- Volume 17 - 2004
- Volume 16 - 2003
- Volume 15 - 2002
- Volume 14 - 2001
- Volume 13 - 2000
- Volume 12 - 1999
- Volume 11 - 1998
- Volume 10 - 1997
- Volume 9 - 1996
- Volume 8 - 1995
- Volume 7 - 1994
- Volume 6 - 1993
- Volume 5 - 1992
- Volume 4 - 1991
- Volume 3 - 1990
- Volume 2 - 1989
- Volume 1 - 1988
Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application
Cited by
Export citation
- BibTex
- RIS
- TXT
@Article{JPDE-37-309,
author = {Li , JingPeng , Weimin and Wang , Yue},
title = {Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application},
journal = {Journal of Partial Differential Equations},
year = {2024},
volume = {37},
number = {3},
pages = {309--325},
abstract = {
We study the regularity and convergence of solutions for the $n$-dimensional ($n=2,3$) fourth-order vector-valued Helmholtz equations
\begin{equation*}\tag{VFHE}\label{VFHE} \mathbf{u}-\beta \Delta \mathbf{u}+\gamma (-\Delta)^{2} \mathbf{u} =\mathbf{v} \end{equation*} for a given $\mathbf{v}$ in several Sobolev spaces, where $ \beta>0$ and $\gamma>0$ are two given constants. By making use of the Fourier multiplier theorem, we establish the regularity and the $L^{p}-L^{q}$ estimates of solutions for Eq. (VFHE) under the condition $\mathbf{v}\in L^{p}(\mathbb{R}^{n})$. We then derive the convergence that a solution $\mathbf{\mathbf{u}}$ of Eq. (VFHE) approaches $\mathbf{v}$ weakly in $L^{p}(\mathbb{R}^{n})$ and strongly in $L^{q}(\mathbb{R}^{n})$ as the parameter pair $( \beta,\gamma)$ approaches $(0,0)$. In particular, as an application of the above results, for $(\mathbf{v},\mathbf{u})$ solving the following viscous incompressible fluid equations
$$\begin{align*}\tag{INS}\label{INS} \begin{cases} \mathbf{v}_{t}+\mathbf{u}\cdot \nabla \mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{u}^{T}+\nabla p= \nu \Delta \mathbf{v},\\ \mbox{div}~\mathbf{v}=\mbox{div}~\mathbf{u}=0, \end{cases} \end{align*}$$
we gain the strong convergence in $L^{\infty}\left([0,T],L^{s}(\mathbb{R}^{n})\right )$ from the Eqs. (VFHE)-(INS) to the Navier-Stokes equations as the parameter pair $(\beta,\gamma)$ tending to $(0,0)$, where $s=\frac{2h}{h-2}$ with $h>n$.
},
issn = {2079-732X},
doi = {https://doi.org/10.4208/jpde.v37.n3.6},
url = {http://global-sci.org/intro/article_detail/jpde/23345.html}
}
TY - JOUR
T1 - Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application
AU - Li , Jing
AU - Peng , Weimin
AU - Wang , Yue
JO - Journal of Partial Differential Equations
VL - 3
SP - 309
EP - 325
PY - 2024
DA - 2024/08
SN - 37
DO - http://doi.org/10.4208/jpde.v37.n3.6
UR - https://global-sci.org/intro/article_detail/jpde/23345.html
KW - Fourier multiplier theorem, fourth-order Helmholtz equation, regularity, convergence.
AB -
We study the regularity and convergence of solutions for the $n$-dimensional ($n=2,3$) fourth-order vector-valued Helmholtz equations
\begin{equation*}\tag{VFHE}\label{VFHE} \mathbf{u}-\beta \Delta \mathbf{u}+\gamma (-\Delta)^{2} \mathbf{u} =\mathbf{v} \end{equation*} for a given $\mathbf{v}$ in several Sobolev spaces, where $ \beta>0$ and $\gamma>0$ are two given constants. By making use of the Fourier multiplier theorem, we establish the regularity and the $L^{p}-L^{q}$ estimates of solutions for Eq. (VFHE) under the condition $\mathbf{v}\in L^{p}(\mathbb{R}^{n})$. We then derive the convergence that a solution $\mathbf{\mathbf{u}}$ of Eq. (VFHE) approaches $\mathbf{v}$ weakly in $L^{p}(\mathbb{R}^{n})$ and strongly in $L^{q}(\mathbb{R}^{n})$ as the parameter pair $( \beta,\gamma)$ approaches $(0,0)$. In particular, as an application of the above results, for $(\mathbf{v},\mathbf{u})$ solving the following viscous incompressible fluid equations
$$\begin{align*}\tag{INS}\label{INS} \begin{cases} \mathbf{v}_{t}+\mathbf{u}\cdot \nabla \mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{u}^{T}+\nabla p= \nu \Delta \mathbf{v},\\ \mbox{div}~\mathbf{v}=\mbox{div}~\mathbf{u}=0, \end{cases} \end{align*}$$
we gain the strong convergence in $L^{\infty}\left([0,T],L^{s}(\mathbb{R}^{n})\right )$ from the Eqs. (VFHE)-(INS) to the Navier-Stokes equations as the parameter pair $(\beta,\gamma)$ tending to $(0,0)$, where $s=\frac{2h}{h-2}$ with $h>n$.
Li , JingPeng , Weimin and Wang , Yue. (2024). Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application.
Journal of Partial Differential Equations. 37 (3).
309-325.
doi:10.4208/jpde.v37.n3.6
Copy to clipboard