# Questions tagged [navier-stokes]

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### 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{ ...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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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