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Zachary Bradshaw

Publications and source records attributed to Zachary Bradshaw.

30 records · Page 2Linked to original sources
arXiv2019

Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

math.AP↗
arXiv2019

Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical $L^2$-based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical $L^2$-based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.

math.AP↗
arXiv2018

An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem

It is shown--within a mathematical framework based on the suitably defined scale of sparseness of the super-level sets of the positive and negative parts of the vorticity components, and in the context of a blow-up-type argument--that the ever-resisting `scaling gap', i.e., the scaling distance between a regularity criterion and a corresponding a priori bound (shortly, a measure of the super-criticality of the 3D Navier-Stokes regularity problem), can be reduced by an algebraic factor; since (independent) fundamental works of Ladyzhenskaya, Prodi and Serrin as well as Kato and Fujita in 1960s, all the reductions have been logarithmic in nature, regardless of the functional set up utilized. More precisely, it is shown that it is possible to obtain an a priori bound that is algebraically better than the energy-level bound, while keeping the corresponding regularity criterion at the same level as all the classical regularity criteria. The mathematics presented was inspired by morphology of the regions of intense vorticity/velocity gradients observed in computational simulations of turbulent flows, as well as by the physics of turbulent cascades and turbulent dissipation.

math.AP↗
arXiv2018

Discretely self-similar solutions to the Navier-Stokes equations with data in $L^2_{\mathrm{loc}}$ satisfying the local energy inequality

Chae and Wolf recently constructed discretely self-similar solutions to the Navier-Stokes equations for any discretely self similar data in $L^2_{\mathrm{loc}}$. Their solutions are in the class of local Leray solutions with projected pressure, and satisfy the "local energy inequality with projected pressure". In this note, for the same class of initial data, we construct discretely self-similar suitable weak solutions to the Navier-Stokes equations that satisfy the classical local energy inequality of Scheffer and Caffarelli-Kohn-Nirenberg. We also obtain an explicit formula for the pressure in terms of the velocity. Our argument involves a new purely local energy estimate for discretely self-similar solutions with data in $L^2_{\mathrm{loc}}$ and an approximation of divergence free, discretely self-similar vector fields in $L^2_{\mathrm{loc}}$ by divergence free, discretely self-similar elements of $L^3_w$.

math.AP↗
arXiv2017

Discretely self-similar solutions to the Navier-Stokes equations with Besov space data

We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space $\dot B^{-1+3/p}_{p,\infty}$ where $3< p< 6$. We also construct discretely self-similar solutions for divergence free initial data in $\dot B^{-1+3/p}_{p,\infty}$ for $3 1$. These results extend those of \cite{BT1} which dealt with initial data in $L^3_w$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$ for $p>3$. We also provide several concrete examples of vector fields in the relevant function spaces.

math.AP↗
arXiv2016

Frequency localized regularity criteria for the 3D Navier-Stokes equations

Two regularity criteria are established to highlight which Littlewood-Paley frequencies play an essential role in possible singularity formation in a Leray-Hopf weak solution to the Navier-Stokes equations in three spatial dimensions. One of these is a frequency localized refinement of known Ladyzhenskaya-Prodi-Serrin-type regularity criteria restricted to a finite window of frequencies the lower bound of which diverges to $+\infty$ as $t$ approaches an initial singular time.

math.AP↗
arXiv2016

Rotationally corrected scaling invariant solutions to the Navier-Stokes equations

We introduce new classes of solutions to the three dimensional Navier-Stokes equations in the whole and half spaces that add rotational correction to self-similar and discretely self-similar solutions. We construct forward solutions in these new classes for arbitrarily large initial data in $L^3_w$ on the whole and half spaces. We also comment on the backward case.

math.AP↗
arXiv2015

Forward discretely self-similar solutions of the Navier-Stokes equations II

For any discretely self-similar, incompressible initial data $v_0$ which satisfies $\|v_0 \|_{L^3_w(\mathbb R^3)}\leq c_0$ where $c_0$ is allowed to be large, we construct a forward discretely self-similar local Leray solution in the sense of Lemarié-Rieusset to the 3D Navier-Stokes equations in the whole space. No further assumptions are imposed on the initial data; in particular, the data is not required to be continuous or locally bounded on $\mathbb R^3\setminus \{0\}$. The same method is used to construct self-similar solutions for any $-1$-homogeneous initial data in $L^3_w$.

math.AP↗
arXiv2013

A note on the surface quasi-geostrophic temperature variance cascade

In this note we examine the dynamical role played by inertial forces on the surface temperature (or buoyancy) variance in strongly rotating, stratified flows with uniform potential vorticity fields. In particular, using a dynamic, multi-scale averaging process, we identify a sufficient condition for the existence of a direct temperature variance cascade across an inertial range. While the result is consistent with the physical and numerical theories of SQG turbulence, the condition which triggers the cascade is more exotic, a fact reflecting the non-locality introduced by fractional dissipation.

math-ph↗
arXiv2013

Blow-up scenarios for 3D NSE exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness

It is shown that, if the vorticity magnitude associated with a (presumed singular) three-dimensional incompressible Navier-Stokes flow blows-up in a manner exhibiting certain {\em time dependent local structure}, then {\em time independent} estimates on the $L^1$ norm of $|ω|\log\sqrt{1+ |ω|^2}$ follow. The implication is that the volume of the region of high vorticity decays at a rate of greater order than a rate connected to the critical scaling of one-dimensional local sparseness and, consequently, the solution becomes sub-critical.

math.AP↗
arXiv2012

A geometric measure-type regularity criterion for solutions to the magnetohydrodynamical system

Several formulations of a local geometric measure-type condition are imposed on super-level sets of mild solutions to the homogeneous incompressible 3D magnetohydrodynamical system with bounded initial data to prevent finite-time singularity formation. Supporting this, results regarding the existence, uniqueness, and real analyticity of mild solutions are established as is a sharp lower bound on the radius of analyticity.

math.AP↗