In this note, we revisit a Liouville-type theorem of Chae and Wolf for stationary Navier-Stokes equations in $\mathbb{R}^3$ [J. Differential Equations 261 (2016) 5541-5560]. We show that their logarithmic improvement of the classical $L^{9/2}$ condition is part of a substantially broader weighted framework. More precisely, we prove that a solution $u\in\dot H^1(\mathbb{R}^3)$ is necessarily trivial whenever $$ \int_{\mathbb{R}^3}|u(x)|^{9/2}\,\omega(|u(x)|)\,dx<+\infty, $$ for every positive, nondecreasing and bounded weight $\omega$ satisfying a mild growth condition near the origin. This structural condition encompasses the logarithmic weight due to Chae and Wolf, as well as a hierarchy of iterated-logarithmic weights and (genuinely) non-logarithmic examples, including a dyadic weight. Our result identifies a broader class of weighted integrability conditions under which the triviality of stationary Navier-Stokes solutions follows, and shows that the mechanism underlying the Chae-Wolf improvement is not intrinsically tied to a specific logarithmic weight or to a single logarithmic scale.
We study the two-dimensional stationary incompressible Navier--Stokes equations on $\mathbb R^2$ with fractional dissipation $(-\Delta)^s$. In the full range $s \in (0,1)$, we prove that every smooth solution satisfying the natural energy condition $u\in\dot{\mathrm H}^s(\mathbb R^2;\mathbb R^2)$ has $u\equiv0$ and con...
Nicola De Nitti, Lukas Niebel, Jia-Qi Yang· 0 citations
We prove Liouville-type theorems for smooth solutions $(u,p)$ of the stationary Navier--Stokes equations in $\mathbb R^3$ satisfying the finite Dirichlet energy condition and the uniform decay condition. Our results give geometric refinements of two recent Osgood-type criteria, namely the relative decay criterion and t...
We investigate positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u = u^{p^*-1} h(u), \qquad 1<p<n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation...
We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate aux...
We prove a sharp nonlinear version of the celebrated Li-Yau inequality for positive solutions of the normalized parabolic $p$-Laplacian equation $u_t = \Delta u + (p-2)|\nabla u|^{-2}\nabla^2u(\nabla u,\nabla u)$, $1<p<\infty$, on a closed Riemannian manifold with nonnegative Ricci curvature, the equation being interpr...
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...
Fan-Ze Kong, Xiao-Yu Zeng· 0 citations
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