Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy

Alexey Cheskidov; Hedong Hou

Home/Archive/arXiv

arXiv · 2610.18797

Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy

Alexey Cheskidov·Hedong Hou

View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove the global existence of solutions to the Navier-Stokes equations for any divergence-free distributional initial data. The solutions are smooth in both time and space at positive times. When the initial data belong to the Sobolev spaces $W^{-1,p}$ for $1<p<\infty$, our constructed solutions are weak in the classical sense. We also characterize the distributional initial data admitting weak solutions and obtain the sharp ranges in Besov and Triebel-Lizorkin spaces.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexey Cheskidov, Hedong Hou. 2026-10-13. Global existence of smooth solutions to the Navier-Stokes equations for initial data with infinite energy. https://arxiv.org/abs/2610.18797

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows

In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $( Ay, 0, Ay )$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$. We show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\|u_{\rm in}\|_{H^2}\leq C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\|v_{\rm in}-(Ay,0,Ay)\|_{H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times\mathbb{R}\times\mathbb{T}$.

math.AP↗

arXiv2026

Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equation in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem \ref{thm1}). Furthermore, by carefully analyzing the asymptotic expansion of the Dirichlet energy $\|\nabla_{\mathbb{B}^2} u\|_{L^2(\mathbb{B}^2)}^2$ as $λ\to 0$, and establishing that \[ \lim_{\substack{λ\to 1/4}} \int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u_λ|^2 \, dV_{\mathbb{B}^2} = +\infty \] for~the ~nonzero~least energy solution, we are able to prove the existence of a critical point for the Moser-Trudinger functional $F(u) = \int_{\mathbb{B}^2} (e^{u^2} - 1) \, dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2} |\nabla_{\mathbb{B}^2} u|^2 \, dV_{\mathbb{B}^2} = Λ$ for $Λ\in (4π, +\infty)$, up to some Möbius transformation. This result is substantially different from the non-existence of critical points for the Moser-Trudinger functional in the Euclidean disk. Finally, by applying a local Pohozaev identity around the origin and exploiting the uniqueness of solutions to the Cauchy initial value problem for ODEs, we prove that the Moser-Trudinger equation has a unique positive solution up to some Möbius transformation when $λ$ is close to $0$.

math.AP↗

arXiv2026

Local asymptotics for singular solutions to critical Hartree equations

For $n\geqslant3$ and $0<α<n$, we study positive classical solutions of critical Hartree equations that blow up at the origin. For the global problem on $\mathbb R^n\setminus\{0\}$, with the Hartree potential integrated over $\mathbb R^n$, we prove that solutions in $L^p(\mathbb R^n)$, where $p=(n+α)/(n-2)$, are radial and nonincreasing in $|x|$. We establish an integral representation and apply moving spheres to this nonlocal problem. For the local problem on $B_R\setminus\{0\}$, with the Hartree potential integrated over $B_R$, we prove $u(x)\leqslant C|x|^{-(n-2)/2}$ near the origin and \begin{equation*} u(x)=\bar u(|x|)\bigl(1+\mathcal O(|x|)\bigr)\qquad\text{as }x\to0, \end{equation*} where $\bar u$ is the spherical average of $u$. A main difficulty is controlling the nonlocal Hartree potential under Kelvin inversion and in the blow-up limit. The local proof combines blow-up analysis and moving spheres with estimates of the integral tails and an auxiliary Liouville lemma. We also give an alternative proof of the classification of positive $C^2$ solutions of the critical Hartree equation in $\mathbb R^n$.

math.AP↗