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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}...

In Ebin and Marsden's paper Groups of Diffeomorphisms and the Motion of an Incompressible Fluid, there is a footnote on the first page indicating that non-homogeneous cases and the case of ...

I am looking into stability for certain nonlinear PDE on $\mathbb{R}$ around a specific steady solution, $f_0(x)$. The nonlinear Cauchy PDE is given by: $\dfrac{\partial f(x,t)}{\partial t}=\mathbf{N}...

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...

I would be interested to know if a certain set of periodic solutions for the two-dimensional Navier-Stokes equations is closed generically. Many similar (yet not identical) set-ups can be found in the ...