Questions tagged [navier-stokes]

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Parabolic regularization for the Navier-Stokes equations

I'm looking for some references about a result on the Navier-Stokes equations which seems to be folklore but for which I didn't manage to find a proof. The setting is the following : Let $Q=\mathbb{R}^...
3
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0answers
292 views

On RH in the Clay Institute list

As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...
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1answer
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How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]

How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ? Is the pressure limited or can it be any amount?
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1answer
91 views

Regularity in Navier Stokes from $L^2$ bound on vorticity

How would one show that if $\omega$ is the vorticity associated to $\partial_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and $$\omega\in L^\infty([0,...
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1answer
158 views

Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$

Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions). Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$. Link to the problem (paper "...
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1answer
153 views

Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$

The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$ Where $$u = [u_1,u_2,\ldots u_n]^T$$ Now I want to rewrite these same equations but with a new ...
64
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3answers
5k views

Should water at the scale of a cell feel more like tar?

The Navier-Stokes equations are as follows, $$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$ where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure. Some elementary ...
4
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0answers
120 views

Existence results for Lagrangian solutions to the Incompressible Euler Equation?

It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
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18 views

Symmetric stationary (divergence free) solution of Navier-Stokes in a $2$-$D$ ball?

Is there an explicit example of symmetric stationary (divergence free) solution of Navier-Stokes in a $2$-$D$ ball?
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81 views

Linearization Navier-Stokes

I am considering the classical form of the stationary Navier-Stokes equation given by $$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...
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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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43 views

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 $\...
2
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1answer
222 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(\...
2
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1answer
102 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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83 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 ...
5
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1answer
259 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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67 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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80 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 \...
5
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1answer
182 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?
3
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1answer
246 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). ...
4
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1answer
285 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 ...
5
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0answers
139 views

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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2answers
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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 ...
3
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1answer
123 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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1answer
90 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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197 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. ...
3
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1answer
294 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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1answer
475 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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1answer
104 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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1answer
100 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_{\...
2
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1answer
145 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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1answer
244 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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0answers
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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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260 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^...