arXiv subjects

Joseph P. Davies

Publications and source records attributed to Joseph P. Davies.

4 recordsLinked to original sources

arXiv2023

Properties of Navier-Stokes mild solutions with initial data in subcritical Lorentz spaces

For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n}) \hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$ ($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply in particular that a local in time mild Navier-Stokes solution cannot become unbounded in the $L^{p,q}(\mathbb{R}^{n})$-norm before it becomes unbounded in the norm of the larger subcritical Besov space $\dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$. In view of the known local theory in such large Besov spaces, this can be thought of as a `propagation of regularity' type of result; here, we provide a self-contained local theory (including scaling-appropriate `blow-up estimates', similar to those established by J. Leray in the Lebesgue setting) in the Lorentz setting, along with uniqueness results which imply such propagation of regularity. Our existence results are based on the method of T. Kato (1984) with Lorentz spaces replacing Lebesgue spaces throughout, and are given without any reference to the Besov framework. The uniqueness results are similarly self-contained, extending certain Lebesgue space methods of Fabes-Jones-Riviere (1972) to the Lorentz setting. We also establish certain continuity properties of the solutions which are constructed. (A more detailed abstract is provided in the article itself.)

math.AP↗

arXiv2023

Properties of Navier-Stokes mild solutions in sub-critical Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in(0,1)}$

We consider mild solutions to the Navier-Stokes initial-value problem which belong to certain ranges $Z_{p,q}^{s}(T,n):=\widetilde{L}^{1}(0,T;\dot{B}_{p,q}^{s+2}(\mathbb{R}^{n}))\cap\widetilde{L}^{\infty}(0,T;\dot{B}_{p,q}^{s}(\mathbb{R}^{n}))$ of Chemin-Lerner spaces. For $n=3$, $ε\in(0,1)$ and $f\in\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})$, Chemin and Gallagher (Tunis. J. Math., 2019) construct a local solution $u\in\cap_{T'\in(0,T_{f,ε}^{*})}Z_{\infty,\infty}^{-1+ε}(T',3)$ with maximal existence time ${T_{f,ε}^{*}\gtrsim_{φ,ε}{\|f\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{3})}^{-2/ε}}$, where $φ$ is the cutoff function used to define the Littlewood-Paley projections. We improve on this result as follows: for $n\geq 1$, $ε\in(0,1)$, $s\in(-1,\infty)$, $p,q\in[1,\infty]$, and initial data $f\in\dot{B}_{p,q}^{s}(\mathbb{R}^{n})\cap\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})$, we prove that there exists a unique local solution $u\in\cap_{T'\in(0,T^*_f)}\left(Z_{p,q}^{s}(T',n)\cap Z_{\infty,\infty}^{-1+ε}(T',n)\right)$ which, along with its maximal existence time $T_{f}^{*}\in(0,\infty]$, is independent of $ε,s,p,q$. If $T_{f}^{*}$ is finite, then we have the blow-up estimate (with explicit dependence on $ε$) ${\|u(t)\|}_{\dot{B}_{\infty,\infty}^{-1+ε}(\mathbb{R}^{n})}\gtrsim_φε(1-ε){(T_{f}^{*}-t)}^{-ε/2}$ for all $t\in(0,T_{f}^{*})$. The solution is unique among all solutions in the larger class $\cap_{T'\in(0,T_{f}^{*})}\cup_{α\in(2,\infty)}L^α(0,T';L^{\infty}(\mathbb{R}^{n}))$, and if $T_{f}^{*}<\infty$ then $u\notin L^{2}(0,T_{f}^{*};L^{\infty}(\mathbb{R}^{n}))$. We also establish additional properties of the solution, depending on the Besov spaces to which the initial data belongs.

math.AP↗

arXiv2022

Navier-Stokes blow-up rates in certain Besov spaces whose regularity exceeds the critical value by $\boldsymbol{ε\in [1,2]}$

For a solution $u$ to the Navier-Stokes equations in spatial dimension $n\geq3$ which blows up at a finite time $T>0$, we prove the blowup estimate ${\|u(t)\|}_{\dot{B}_{p,q}^{s_{p}+ε}(\mathbb{R}^n)}\gtrsim_{φ,ε,(p\vee q\vee 2)}{(T-t)}^{-ε/2}$ for all $ε\in[1,2)$ and $p,q\in[1,\frac{n}{2-ε})$, where $s_{p}:=-1+\frac{n}{p}$ is the scaling-critical regularity, and $φ$ is the cutoff function used to define the Littlewood-Paley projections. For $ε=2$, we prove the same type of estimate but only for $q=1$: ${\|u(t)\|}_{\dot{B}_{p,1}^{s_{p}+2}(\mathbb{R}^n)}\gtrsim_{φ,(p\vee 2)}{(T-t)}^{-1}$ for all $p\in [1,\infty)$. Under the additional restriction that $p,q\in[1,2]$ and $n=3$, these blowup estimates are implied by those first proved by Robinson, Sadowski and Silva (J. Math. Phys., 2012) for $p=q=2$ in the case $ε\in(1,2)$, and by McCormick, Olson, Robinson, Rodrigo, Vidal-López and Zhou (SIAM J. Math. Anal., 2016) for $p=2$ in the cases $(ε,q)=(1,2)$ and $(ε,q)=(2,1)$.

math.AP↗

arXiv2022

Notions of solution and weak-strong uniqueness criteria for the Navier-Stokes equations in Lorentz spaces

For initial data $f\in L^{2}(\mathbb{R}^n)$ ($n\geq 2$), we prove that if $p\in(n,\infty]$, any solution $u\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}\cap L_{t}^{\frac{2p}{p-n}}L_{x}^{p,\infty}$ to the Navier-Stokes equations satisfies the energy equality, and that such a solution $u$ is unique among all solutions $v\in L_{t}^{\infty}L_{x}^{2}\cap L_{t}^{2}H_{x}^{1}$ satisfying the energy inequality. This extends well-known results due to G. Prodi (1959) and J. Serrin (1963), which treated the Lebesgue space $L_{x}^{p}$ rather than the larger Lorentz (and `weak Lebesgue') space $L_{x}^{p,\infty}$. In doing so, we also prove the equivalence of various notions of solutions in $L_{x}^{p,\infty}$, generalizing in particular a result proved for the Lebesgue setting in Fabes-Jones-Riviere (1972).

math.AP↗