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Volume 37, Issue 3
Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application

Jing Li, Weimin Peng & Yue Wang

DOI: 10.4208/jpde.v37.n3.6

J. Part. Diff. Eq., 37 (2024), pp. 309-325.

Published online: 2024-08

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  • 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 $h>n$.
  • Keywords

Fourier multiplier theorem, fourth-order Helmholtz equation, regularity, convergence.

  • AMS Subject Headings

35J05, 35J35

  • Copyright

COPYRIGHT: © Global Science Press

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@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
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