For $n\geq5$, let $(u,p)$ be a smooth solution of the stationary incompressible Navier-Stokes equations in $\R^n\setminus\{x'=0\}$, where $\{x'=0\}$ is the $x_n$-axis. We prove that the scale-invariant transverse bound $|u(x)|\leq C |x'|^{-1}$ forces $u\equiv0$. This result relies on a quantitative local regularity theorem, which shows that the transverse bound condition yields uniform control of $u$ and $\nabla u$ in a smaller ball and removes the possible singularity along the axis. The proof combines weak extension across the axis, approximate Green functions for an adjoint drift operator, a one-sided bound for the total head pressure, a localized Frehse-R\r{u}\v{z}i\v{c}ka weighted estimate, and a finite Stokes-Morrey bootstrap. Applying the local theorem at arbitrarily large scales yields whole-space rigidity and removability of line singularities. This together with the asymptotic expansions on the exterior domains in the case $|u(x)|\leq C |x|^{-1}$ in [1], yields the same expansion in exterior domains under the condition $|u(x)|\leq C |x'|^{-1}$.
For every integer $n\ge4$, we prove the existence of a $(-1)$-homogeneous solution of the stationary incompressible Navier-Stokes equations on $\mathbb{R}^n\setminus\{0\}$ with an arbitrary locally Lipschitz $(-3)$-homogeneous force. The pressure is $(-2)$-homogeneous, and the solution is regular away from the origin....
We prove a critical Morrey rigidity theorem for the five-dimensional stationary Navier--Stokes equations. More precisely, every smooth solution on $\mathbb R^5\setminus\{0\}$ satisfying \[ \sup_{R>0}R^{-2}\int_{B_R}|u|^3\,dx<\infty \] is identically zero, up to an additive constant in the pressure. This replaces the po...
Yu-Bo Chen, Wen-Dong Wang, Xiao Wang et al.· 2 citations
This paper and its companion Part II prove that the incompressible Euler equations in the plane, without external force, admit two distinct renormalized weak solutions with the same initial vorticity $f$, where $f$ is radial and nonnegative and lies in $(L^1\cap L^q)(\mathbb{R}^2)$ for every $q<\infty$ but is not bound...
For $n\ge3$ and $\gamma>-1$, we study normal solutions of \begin{equation*} (-\Delta)^{n/2}u=(n-1)!\,|x|^{n\gamma}e^{nu} \quad\hbox{in }\R^n. \end{equation*} We reformulate this problem on the cylinder via the Emden--Fowler transform. We first establish Kelvin symmetry for normal solutions. We then prove existence and...
Hao-Yu Li, Tuo Li, Jun-Cheng Wei et al.· 0 citations
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
Recently, the existence of regular solutions $(u,p)$ to the stationary incompressible Navier--Stokes equations on $\mathbb R^n$ with bounded compactly supported forces $f$ for $5\le n\le15$ was established by Li--Yang (Comm. Math. Phys., 2022). However, it is still unknown whether there exists a regular solution for di...
Wen-Dong Wang, Guo-Xu Yang· 2 citations· ⚡2
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