arXiv subjects

Joel Avrin

Publications and source records attributed to Joel Avrin.

2 recordsLinked to original sources

arXiv2019

The 3-D Spectrally-Hyperviscous Navier-Stokes Equations on Bounded Domains with Zero Boundary Conditions

We develop a mathematically and physically sound definition of the spectrally-hyperviscous Navier-Stokes equations (SHNSE) on general bounded domains Ωwith zero (no-slip) boundary conditions prescribed on \varGamma=\partial\varOmega. Previous successful studies of the SHNSE have been limited to periodic-box domains. There are significant theoretical obstacles to overcome in extending the SHNSE beyond this case. We solve them with the help of the Helmholtz decomposition, and with our new formulation of the SHNSE on general bounded domains in hand we then establish foundational results, beginning with the existence of globally regular solutions. Given that the SHNSE is meant to approximate the NSE for small μor large m, we establish this rigorously in the general case by adapting the weak subsequence convergence results of [5] to hold here. On intervals [0,T] with a common H^{1}-bound we deepen this sense of approximation by obtaining strong convergence. First, by using estimates depending only on the common H^{1}-bound to maximize computational applicability we show that SHNSE solutions converge uniformly in H^{1} to the NSE solution as either μ\rightarrow0 or m\rightarrow\infty. Then in cases in which bootstrapped higher-order bounds can be readily used we show that higher-order convergence results hold. Our final results use the Stokes-pressure methodology developed in [29], [30] to recast the SHNSE in a form which like the NSE reformulation in [29], [30] is more adaptable to computation and the specification of boundary values for the pressure.

math.AP↗

arXiv2014

Regularity Criteria of BKM type in Distributional Spaces for the 3-D Navier-Stokes Equations on Bounded Domains

In the classic work of Beale-Kato-Majda ({[}2{]}) for the Euler equations in $\mathbb{R^{\mathrm{3}}}$, regularity of a solution throughout a given interval $[0,T_{*}]$ is obtained provided that the curl $ω$ satisfies $ω\in L^{1}((0,T);L^{\infty}(\mathbb{R^{\textrm{\ensuremath{3}}}})$ for all $T<T_{*}$, and the arguments apply equally well to the Navier-Stokes equations (NSE) in $\mathbb{R^{\mathrm{3}}}$. The spatial $L^{\infty}$-criterion imposed on the curl was generalized to other function spaces by various authors ({[}9{]}, {[}10{]}, {[}11{]}). In {[}8{]} regularity results of this type are obtained on localized balls. In this paper for the NSE case and on general bounded domains $Ω$ in $\mathbb{R^{\mathrm{3}}}$, we obtain a regularity result of BKM type that allows $ω$ to be a distribution. Specifically, we show that if $u$ is a Leray solution of the 3-D NSE on the interval $(0,T)$ and if $ω\in L^{s}((0,T);H^{-1,p}(Ω))$ where $\frac{2}{s}+\frac{3}{p}=1$ for some $p\in(3,\infty]$, then $u$ is a regular solution on $\left(0,T]\right)$; in particular for $p=\infty$ we have a regular solution when $ω\in L^{2}((0,T);H^{-1,\infty}(Ω))$, which directly strengthens the results in {[}2{]} by one order of (negative) derivative in terms of the spatial criteria for regularity. Our results thus impose more stringent conditions on time than the BKM results and their generalizations described above, but as far as we are aware the results here represent the first of BKM type for the NSE that allow $ω$ to spatially be a distribution.

math.AP↗