Regularity and Convergence for the Fourth-Order Helmholtz Equations and an Application (Journal of Partial Differential Equations, 2024)
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$$\begin{align*}\tag{INS}\label{INS} \begin{cases} \mathbf{v}_{t}+\mathbf{u}\cdot abla \mathbf{v}+\mathbf{v}\cdot abla \mathbf{u}^{T}+ abla p= u  \Delta \mathbf{v},\\ \mbox{div}~\mathbf{v}=\mbox{div}~\mathbf{u}=0, \end{cases} \end{align*}$$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 $h>n$.