Questions tagged [navier-stokes]
The navier-stokes tag has no usage guidance.
59 questions
0
votes
0
answers
51
views
Time periodic Euler flows
What are some examples of solutions to the incompressible Euler equation on the torus $u:\mathbb{R}\times \mathbb{T}^d\rightarrow \mathbb{R}$ (with $d\in \{2,3\}$)
$$\partial_t u+u\cdot \nabla u +\...
- 94
4
votes
1
answer
167
views
Pressureless explicit solutions to incompressible Euler
What are some examples of (semi-)explicit solutions of the incompressible Euler equations which satisfy the following
they are pressureless
they are periodic in space
they have nontrivial time ...
- 94
2
votes
1
answer
160
views
Is there a name for this nice property of the usual weak incompressible Navier-Stokes equation?
Navier-Stokes is a non-linear PDE, and there is no standard, general theory of weak solutions for nonlinear PDEs. But the literature on weak solutions to the incompressible Navier-Stokes constantly ...
- 11.1k
12
votes
1
answer
450
views
Can Buckmaster-Vicol paradoxical solutions to Navier Stokes show macroscopic motion?
Villani, in his paper "Paradoxe de Scheffer-Shnirelman ..." (MR2648676, Zbl 1404.35338), describes the paradoxical solutions to the Euler equation by Scheffer and Shnirelman, in the form ...
- 11.1k
2
votes
0
answers
94
views
Non-selfadjoint operators and physical systems
There are plenty of examples of non-selfadjoint operators modelizing physical phenomena: to name a few, let's quote the the heat equation (\ref{HEAT}, see below), the Navier-Stokes system for ...
- 16.2k
0
votes
0
answers
55
views
Question on the modelling of (viscous) fluid in a bag with holes
Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture)
What is the corresponding PDE to model the ...
- 1,334
2
votes
0
answers
164
views
$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)
This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
- 3,477
2
votes
0
answers
98
views
Are analytic solutions for the Navier-Stokes equations sufficient?
Generally, we ask for solutions of the Navier-Stokes equations, when the starting conditions are in the Schwartz space.
However, I am wondering, whether it is possible to consider just analytic ...
- 749
4
votes
1
answer
210
views
Is Stokes equation a saddle point problem or a minimum problem?
Consider $\Omega \subset \mathbb{R}^2 $ (or $\mathbb{R}^3$). The well known stationary Stokes equations in the incompressible case are
\begin{equation}
\begin{cases}
- \Delta u + \nabla p = f \text{ ...
- 697
2
votes
1
answer
228
views
The physical meaning of the $L^2$ norm of the gradient of the velocity in N-S equation
I'm reading Asymptotic Properties of Steady Plane Solutions
of the Navier-Stokes Equations
with Bounded Dirichlet Integral written by D. GILBARG and H. F. WEINBERGER. They studied the properties of an ...
- 809
1
vote
1
answer
88
views
Stochastic Stokes flow: where to start from?
I would need to get acquainted to the subject of stochastic Stokes flows, so studying Stokes equations under some noise of some kind, let's say an additive white noise to begin with.
The problem is ...
- 697
2
votes
0
answers
35
views
How to estimate of the pressure in the forced Stokes Equation in $L^1(\Omega)$?
Let's consider the following system
\begin{equation*}
\left\{
\begin{split}
u_t+\nabla P &=& \Delta u+f, \quad &x\in \Omega\times(0,T),\\
\nabla \cdot u&=& 0, & x\in \...
- 21
4
votes
0
answers
98
views
Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?
Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider
\begin{align*}
\begin{cases}
u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
- 313
1
vote
0
answers
86
views
Treating 2D NSE with an $L^4$ contraction mapping
For divergence-free initial data $u_0 \in L^2(\mathbb{T}^2)$, the two-dimensional Navier Stokes equation is known to have a global mild solution. This fact is classical. However, a written-out proof ...
- 96
0
votes
1
answer
1k
views
The derivation of Reynolds-averaged Navier-Stokes equations
The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations)
When we talk about turbulent flows we can represent the velocity of the fluid as:
$$
...
- 1
-1
votes
1
answer
144
views
One or all R^3 Navier-Stokes solutions?
I was just wondering, reading Navier-Stokes official problem description (https://www.claymath.org/sites/default/files/navierstokes.pdf), if only one solution on $R^3$, respecting all required ...
- 9
2
votes
1
answer
148
views
Critical Reynolds numbers for turbulence in 3D and 2D planar Couette flows
In 3 spatial dimensions, the incompressible Navier-Stokes equations are:
$$
\begin{split}
\frac{\partial u_i}{\partial t} + \sum_{j=1}^3 u_j \frac{\partial u_i}{\partial x_j} &= - \frac{\partial p}...
- 1,547
3
votes
0
answers
73
views
Does the incompressible Navier-Stokes equation have a smooth solution if the initial vorticity is smooth and $p$ integrable for any $p$?
Consider the incompressible Navier-Stokes equation on $(0,T)\times \mathbb{R}^3$ for fixed $T>0$.
If the sequence of mollified initial vorticities $(\omega_0^{\nu})_{\nu}$ is uniformly bounded in $...
- 173
1
vote
0
answers
97
views
Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$
I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
- 452
3
votes
0
answers
108
views
Local strong solution to Navier-Stokes equation by Galerkin method without using the eigenfunctions of the Stokes operator
The existence of a local-in-time strong solution to the Navier-Stokes equations in a smoothly bounded domain $\Omega \subseteq \mathbb{R}^3$ (say with no-slip boundary condition) can be obtained by a ...
- 109
30
votes
5
answers
3k
views
Looking for an interesting result on the Navier-Stokes equations
I am in my second year of master in Mathematics and one of my courses consists of a reading of Navier-Stokes Equations by Roger Temam. We have proven the existence for the weak Stokes and Navier-...
- 452
5
votes
0
answers
304
views
Similarity in Navier-Stokes equation and convolution in finite abelian groups?
Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables.
Set for $g \in G$:
$$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
- 3,032
3
votes
0
answers
172
views
Radial-energy decomposition of a velocity field in 2D: is anyone able to show that the lemma below is true (…or false)?
In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma.
Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity
$$\omega=\...
- 1,065
3
votes
1
answer
307
views
A simple question on the Navier-Stokes system
The Navier-Stokes system for incompressible fluids in $\mathbb R^3$
reads as
\begin{align}
&\frac{\partial v}{\partial t}+\mathbb P\bigl((v\cdot \nabla) v\bigr)-\nu \Delta v=0, \quad \text{div} v=...
- 16.2k
3
votes
0
answers
182
views
Parabolic regularization for the Navier-Stokes equations
I'm looking for some references about a result on the Navier-Stokes equations which seems to be folklore but for which I didn't manage to find a proof. The setting is the following :
Let $Q=\mathbb{R}^...
- 161
4
votes
0
answers
518
views
On RH in the Clay Institute list
As everybody knows, the Riemann Hypothesis is one of the problems of the millenium raised by the Clay Institute. Looking at the "official formulation" of various problems, say for instance ...
- 16.2k
-1
votes
1
answer
175
views
How do we approximate the pressure in the Boussinesq equations of hydrodynamics? [closed]
How do we approximate the pressure or the gradient of it in the Boussinesq equations of hydrodynamics ?
Is the pressure limited or can it be any amount?
- 11
1
vote
1
answer
152
views
Regularity in Navier Stokes from $L^2$ bound on vorticity
How would one show that if $\omega$ is the vorticity associated to $\partial_t u+u\cdot \nabla u -\nu \Delta u +\nabla p=0$ (with smooth, compactly supported initial data) and
$$\omega\in L^\infty([0,...
- 94
0
votes
2
answers
241
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 "...
- 698
-3
votes
1
answer
200
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 ...
- 59
67
votes
3
answers
6k
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 ...
- 989
4
votes
0
answers
141
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 ...
- 989
0
votes
0
answers
120
views
Linearization Navier-Stokes
I am considering the classical form of the stationary Navier-Stokes equation given by
$$\frac{1}{Re} (\nabla v, \nabla \phi) + ((v \cdot \nabla) v, \phi) - (p, \nabla \cdot \phi) = (f,\phi);\qquad (\...
- 11
0
votes
0
answers
151
views
Inf sup condition for discrete Stokes problem
I am considering the discrete inf-sup condition for the discrete Stokes problem
$ \inf_{q \in Q_h} \sup_{v \in V_h} \frac{(q, \nabla \cdot v)}{\| q \| \| \nabla v\|} \geq \beta > 0$
This means ...
- 11
1
vote
0
answers
73
views
Density gradient of Navier-Stokes equations in perforated domain
Question: Is it possible to bound the density gradient $\nabla \rho_\epsilon$ of strong solutions to the compressible NSE uniformly in $L^\gamma$?
Preliminaries: Consider a bounded connected domain $\...
- 41
2
votes
1
answer
407
views
Leray projector in $L^{\infty}$ and negative order Besov spaces for the Navier-Stokes equations
I was reading the paper "Norm inflation for the generalized Navier-Stokes equations" which can be found here: https://arxiv.org/abs/1212.3801.
In Lemma 2.1, the authors said for any $\phi \in L^{\...
- 43
0
votes
0
answers
31
views
About parametrization of the interface of a fluid
In Navier Stokes Equation, more precisly in the evolution of a fluid interface I have an extension of the function $u$ such that:
$$u(x,t)=\frac{1}{2\pi} \operatorname{P.V.}\int_\Omega\frac{(x-z(\...
- 31
2
votes
1
answer
210
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^...
- 43
1
vote
0
answers
123
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 ...
- 81
5
votes
1
answer
318
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.
...
- 3,085
3
votes
0
answers
88
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: $$\omega_t ...
- 31
3
votes
0
answers
104
views
Green's function of time-dependent Stokes equation
It is well known that the Green's function of a standard parabolic equation in a bounded domain, say
\begin{align}
\partial_tG(x,y,t)-\Delta_x G(x,y,t)&=0
&&\mbox{for}\,\,\,(x,t)\in \...
- 393
5
votes
1
answer
424
views
Incompressible Navier-Stokes equation with heat conduction
How does the incompressible Navier-Stokes system read with heat conduction?
Where can I find an existence result for its weak solutions?
user123456
3
votes
1
answer
491
views
Difference between linear and parabolic velocity profiles in Stokes flows of two fluids
For droplet interactions in low-Reynolds number flow, solutions are available when the underlying flow can be written as linear compositions of strain and rotation, see Batchelor & Green (1972a).
...
- 41
6
votes
1
answer
432
views
Stationary Navier-Stokes solutions
Are there known nontrivial ($u\neq0$) stationary solutions to Navier-Stokes equations in $\mathbb R^3$ ? Not square integrable of course (that's impossible), but with self-similar amplitudes of ...
- 3,085
5
votes
0
answers
176
views
Estimate of $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$ of Navier Stokes equations
My question is how to estimate the term $\Vert \nabla u \Vert_{L^{\infty}(\Omega)}$. Here we consider the 2D incompressible Navier Stokes equations:$$u_t -\Delta u+u\cdot \nabla u+\nabla p=f$$ and $$\...
- 83
13
votes
2
answers
3k
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 ...
- 233
3
votes
1
answer
226
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 ...
- 115
1
vote
1
answer
116
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?
- 11
3
votes
0
answers
349
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.
...
- 139