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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 \(u(t=0,\cdot)=u_0(\cdot)\).

Claim: The pressure \(p\) can be interpreted as Lagarange multiplicator that comes from incompressibility constraint \(\nabla \cdot u=0\).

Basis: Lagrange multiplicator are associated to variational problem.

Problem: Navier-Stokes equation has no known variational structure.

Alternative: Consider special cases: Stokes and Euler equation, which have a variational structure.

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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 \(L^1(\Omega)\).

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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 \(u:\R_+\times \R^N \to \R\). The prefactor \(\frac{1}{s(t)^N}\) ensures that \(u\) conserves mass, i.e.
1 = \int_{\R^N} \varrho(x,t) \; \dx{x} \stackrel{x\mapsto ys}{=} \int_{\R^N} \frac{1}{s(t)^N} \varrho(y s(t),t)\;\dx{y} = \int_{\R^N} u(y,\tau(t)) \; \dx{y}.
\] Continue reading

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 \(\R^n\). The partial differential equation is in divergence form and conserves mass. Hence, also \(\varrho(\cdot,t)\) is a probability density on \(\R^n\). In the case, where \(H\) has some growth at \(\infty\), the equilibrium solutions \(\varrho_\infty:\R^n \to \R\) are characterized by
\[ \beta \nabla \varrho_\infty + \varrho_\infty \nabla H =0 \] leading to the solution
\[ \varrho_\infty(x) = Z^{-1} e^{-\beta H(x)} \quad \text{with} \quad Z = \int e^{-\beta H(x)} \; \dx x \] Continue reading