Questions tagged [navier-stokes]

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Inf sup condition for discrete Stokes problem

I am considering the discrete inf-sup condition for the discrete Stokes problem $ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$ This means ...
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Density gradient of Navier-Stokes equations in perforated domain

Question: Is it possible to bound the density gradient $\nabla \rho_\epsilon$ of strong solutions to the compressible NSE uniformly in $L^\gamma$? Preliminaries: Consider a bounded connected domain $\...
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1answer
129 views

Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations

I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801. In Lemma 2.1, the authors said for any $\phi \in L^{\...
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About parametrization of the interface of a fluid

In Navier Stokes Equation, more precisly in the evolution of a fluid interface I have an extension of the function $u$ such that: $$u(x,t)=\frac{1}{2\pi} \operatorname{P.V.}\int_\Omega\frac{(x-z(\...
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93 views

Cylindrical coordinates in axis symmetric flow

I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations $$u_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$ We consider in cylindrical coordinates $u=(u^r, u^...
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71 views

2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
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243 views

Obstruction to Navier-Stokes blowup with cylindrical symmetry

Is there a known obstruction to cylindrically symmetric solutions (with swirl) of incompressible 3D Navier-Stokes blowing up in finite time ? EDIT: in the whole space $\mathbb R^3$, I forgot to say. ...
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55 views

Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain

Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
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75 views

Green's function of time-dependent Stokes equation

It is well known that the Green's function of a standard parabolic equation in a bounded domain, say \begin{align} \partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0 &&\mbox{for}\,\,\,(x,t)\in \...
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148 views

Incompressible Navier-Stokes equation with heat conduction

How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
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122 views

Difference between linear and parabolic velocity profiles in Stokes flows of two fluids

For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a). ...
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253 views

Stationary Navier-Stokes solutions

Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
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Estimate of $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$ of Navier Stokes equations

My question is how to estimate the term $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$. Here we consider the 2D incompressible Navier Stokes equations:$$u_t -\Delta u+u\cdot \nabla u+\nabla p=f$$ and $$\...
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961 views

Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?

This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold: https://arxiv.org/abs/1709.10033 What's the current ...
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104 views

A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?

Let $u_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{ \begin{array}{ccc} \partial _t ...
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81 views

Vorticity equation for generalized Naiver Stokes equations

In $3D$ or $2D$, I can get the vorticity equation for the incompressible NSE; however, what's the vorticity equation for the generalized NSE? Does the fractional laplacian commute with the curl?
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178 views

Critical spaces and energy estimate in NS equation [closed]

There is a ‘rescaling transformation’ that is particularly significant for the Navier–Stokes equations when they are posed on the whole space, but is also important in the local regularity theory. ...
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204 views

Hadamard-Rybczynski problem

HR problem deals with a spherical fluid viscous drop falling in a different fluid under influence of gravity. The outer fluid is of uniform speed $U$ in direction of gravity, far away from the drop. ...
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353 views

Naviers Stokes equation and machine learning

I am looking for a reference explaining how to solve Navier-Stokes numerically using Machine learning algorithms . Thank you in advance for your help .
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98 views

Mild solution of 2D surface quasi-geostrophic (SQG) equation

I was reading one of Kato's papers on Navier-stokes equations. A mild solution can be denoted as $u= e^{t\Delta}u_0 + \int_{0}^{t} e^{(t-s)\Delta} \mathbb P\nabla \cdot(u \otimes u)ds$, where $\mathbb ...
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87 views

Bound of solutions by initial value of Navier Stokes equations

For the mollified Navier Stokes equations: $$\partial_t u_{\epsilon} - \Delta u_{\epsilon} + \mathbb P \nabla \cdot((u_{\epsilon} \ast \omega_{\epsilon})\otimes u_{\epsilon})=0 $$ $$\nabla \cdot u_{\...
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139 views

Fourier multiplier and Navier Stokes equations

Let $\mu$ be in $\mathcal D (\mathbb R^d)$ with $\mu \geq 0$, i.e. $\mu$ is a test function. Furthermore, we assume $\mu (\xi) =1$ when $|\xi|<1$ and $\mu (\xi) =0$ when $|\xi| \geq 2$. Why is the ...
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215 views

A detail in Kato's paper (Strong $L^p$-Solutions of the Navier-Stokes Equation in $\mathbb R^m$, with Applications to Weak Solutions)

A detail in Kato's paper (Strong $L^p$-Solutions of the Navier-Stokes Equation in $\mathbb R^m$, with Applications to Weak Solution). Here is the link: http://junon.u-3mrs.fr/monniaux/K84.pdf In the ...
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71 views

Books on turbulent compressible fluid (gas) in heated channel

It's been a while I got my hands dirty with simulation of hydrodynamics and it was mostly incompressible and laminar. Now, I need to model turbulent flow in channel with additional external heating ...
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245 views

Dual space of vector fields with null divergence

Let us consider $\mathscr D(\mathbb R^3;\mathbb R^3)$ the space of smooth compactly supported vector fields in $\mathbb R^3$ and let us define $\widetilde{\mathscr D}=\mathscr D(\mathbb R^3;\mathbb R^...