\(\newcommand{\R}{\mathbb{R}}\)

[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

# Category Archives: Uncategorized

# 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

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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

### Description

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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