Questions tagged [navier-stokes]
The navier-stokes tag has no usage guidance.
35
questions
3
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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
votes
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 ...
-1
votes
1answer
62 views
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?
0
votes
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
$$\omega\in L^\infty([0,...
1
vote
1answer
158 views
Band limited initial data : regularity for Navier–Stokes equation defined on a torus $\mathbb{T}^m$
Consider the Navier–Stokes equation and the Euler equation defined on a torus (periodic solutions).
Let the dimensionality of the space $\mathbb{T}^m$ be $m\ge 3$.
Link to the problem (paper "...
-3
votes
1answer
153 views
Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The Euler equations are given as $$ \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$$
Now I want to rewrite these same equations but with a new ...
64
votes
3answers
5k views
Should water at the scale of a cell feel more like tar?
The Navier-Stokes equations are as follows,
$$\dot{u}+(u\cdot \nabla ) u +\nu \nabla^2 u =\nabla p$$
where $u$ is the velocity field, $\nu$ is the viscosity, and $p$ is the pressure.
Some elementary ...
4
votes
0answers
120 views
Existence results for Lagrangian solutions to the Incompressible Euler Equation?
It is known that if a function (which we shall call the lagrangian flow, or lagrangian trajectory) $$X:(\mathbb{R}/\mathbb{Z})^3 \times [0,T] \to \mathbb{R}^3$$ with $X \in H^1_t$ (i.e. has weak time ...
0
votes
0answers
18 views
Symmetric stationary (divergence free) solution of Navier-Stokes in a $2$-$D$ ball?
Is there an explicit example of symmetric stationary (divergence free) solution of Navier-Stokes in a $2$-$D$ ball?
0
votes
0answers
81 views
Linearization Navier-Stokes
I am considering the classical form of the stationary Navier-Stokes equation given by
$$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...
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64 views
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 ...
1
vote
0answers
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
votes
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^{\...
0
votes
0answers
17 views
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:
$$u(x,t)=\frac{1}{2\pi} \operatorname{P.V.}\int_\Omega\frac{(x-z(\...
2
votes
1answer
102 views
Cylindrical coordinates in axis symmetric flow
I am getting stuck in a detail in a paper. It's about the axi symmetric Navier Stokes equations
$$u_t - \nu\,\Delta u + u\cdot \nabla u + \nabla p=0$$
We consider in cylindrical coordinates $u=(u^r, u^...
1
vote
0answers
83 views
2D Stochastic Navier Stokes equations with Navier boundary condition
For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
5
votes
1answer
259 views
Obstruction to Navier-Stokes blowup with cylindrical symmetry
Is there a known obstruction to cylindrically symmetric solutions (with swirl) of incompressible 3D Navier-Stokes blowing up in finite time ?
EDIT: in the whole space $\mathbb R^3$, I forgot to say.
...
2
votes
0answers
67 views
Parabolic regularity for the 2D Navier-Stokes equations in a bounded domain
Suppose we consider 2D Navier-Stokes equations in a bounded domain $\Omega \subseteq \mathbb R^2$, together with suitable boundary conditions so that we can consider the vorticity equation: $$\omega_t ...
3
votes
0answers
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
votes
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
votes
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
votes
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
votes
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 $$\...
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votes
2answers
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Has it been proved that weak solutions to the Navier-Stokes equations are non-unique, and does this prove that the Navier-Stokes are not valid?
This preprint claims that, for finite kinetic energy initial solutions, uniqueness of weak solutions to the Navier-Stokes equations doesn't hold:
https://arxiv.org/abs/1709.10033
What's the current ...
3
votes
1answer
123 views
A solution to the Navier-Stokes equation that is defined for on $[0,T]$ with $T$ large is global?
Let $u_0 \in \dot{H}^{1/2}(\mathbb{R}^3)$. The Fujita-Kato theorem gives rise to a local unique solution $(t,x) \mapsto u(t,x)$ to the Navier-Stokes equations $$\left\{
\begin{array}{ccc}
\partial _t ...
1
vote
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?
3
votes
0answers
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
votes
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. ...
2
votes
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 .
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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 ...
3
votes
0answers
74 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 ...
2
votes
0answers
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^...
