# Pressure as Lagrange multiplicator in Stokes and Euler equation

$$\newcommand{\R}{\mathbb{R}}$$
[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

# weak L¹-convergence

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

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

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