Category Archives: Uncategorized

Pressure as Lagrange multiplicator in Stokes and Euler equation

[stextbox id=”Theorem” caption=”Navier-Stokes equation in \(d=2,3\)”] \[ \begin{aligned}
u &: [0,\infty) \times \Omega \to \R^d &\text{velocity (unknown)} & \text{ length / time } \\
p &: [0,\infty) \times \Omega \to \R^d &\text{pressure (unknown)} & \text{ force / area } \\
f &: [0,\infty) \times \Omega \to \R^d &\text{external volume force (data)} & \text{ force / volume } \\
\varrho &\in [0,\infty) &\text{mass density (data)} & \text{ mass / volume } \\
\mu&\in (0,\infty) &\text{dynamic viscosity (data)} & \text{ pressure \(\cdot\) time }
\end{aligned}\] \[\begin{aligned}
\varrho \left( \partial_t u + u \cdot \nabla u \right) – \mu \Delta u + \nabla p &= f &&\text{in \((0,\infty)\times \Omega\)},\\
\nabla \cdot u &= 0 &&\text{in \((0,\infty)\times \Omega\)},\\
u &= 0 &&\text{on \((0,\infty)\times \partial \Omega\)}
\end{aligned}\] and initial conditions \(u(t=0,\cdot)=u_0(\cdot)\).… Read the rest

weak L¹-convergence

\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)
Let \(\Omega\subset \mathbb{R}^n\), a sequence \((u_n)\) converges weakly to \(u\in L^p(\Omega)\) if
\[ \int_\Omega u_n v\;\dx{x} = \int_\Omega u v \;\dx{x} , \qquad \forall v\in L^\infty(\Omega) . \] As usual the convergence is denoted by \(u_n \rightharpoonup u\) in \(L^1(\Omega)\).

Classic results for porous medium equation

Selfsimilar solutions

\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)
Let us seek for solutions of \(\partial \varrho = \Delta \varrho^m\) satisfying the scaling hypothesis
(x,t)\mapsto (y,\tau):=\left(\frac{x}{s(t)},\tau(t)\right) \qquad \varrho(x,t) = \frac{1}{s(t)^N} u\left(\frac{x}{s(t)},\tau(t)\right),
\] where \(s: \R_+ \to \R_+\) and \(\tau:\R_+ \to \R_+\) are reparametrization of time and \(u:\R_+\times \R^N \to \R\).… Read the rest

Fokker Planck equation


\( \newcommand{\R}{\mathbb{R}} \newcommand{\dx}{\mathrm{d}} \)
The Fokker Planck equation has the form
\partial_t \varrho(x,t) = \nabla \cdot \bigl(\beta^{-1} \nabla \varrho(x,t) + \varrho(x,t) \nabla H(x)\bigr) ,\qquad \varrho(x,0)=\varrho_0(x) ,
\] where \(H:\R^n\to \R\) is a smooth function, \(\beta>0\) some parameter and \(\varrho_0\) a probability density on \(\R^n\).… Read the rest