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4

In general when $f: \Bbb{R}^n \to \Bbb{R}$ is the euclidean norm, defined by \begin{equation} f(x) = |x| = \sqrt{\sum_{i=1}^n x_i^2} \end{equation} then away from the origin, $f$ is $C^{\infty}$ and its derivative is given by \begin{equation} f'(x)(\eta) = \left\langle \dfrac{x}{|x|}, \eta \right\rangle \end{equation} (for all $x \neq 0$, and for all $\eta \...


2

Briefly, the derivative $E'(u)$ is a linear functional on an appropriate function space. Let $v\in C^\infty_c(\mathbb{R})$, then $$ E'(u)(v)=\lim_{t\to 0}\frac{E(u+tv)-E(u)}{t} $$ So \begin{align*} E'(u)(v)&=\int_{\mathbb{R}} [2u(x,t)v(x,t)+2u_x(x,t)v_x(x,t)]\,\mathrm{d}x\\ &=\int_{\mathbb{R}} 2(u(x,t)-u_{xx}(x,t))v(x,t)\,\mathrm{d}x \end{align*} So ...


2

The PDE $u_t + u_x = f-u$ is a non-homogeneous linear advection equation. Let us solve the initial value problem $u(x,0)=u_0(x)$ by using the method of characteristics. We introduce the parametrization $x(t)$ of $x$ such that the total time-derivative of $u(x(t),t)$ satisfies $$ \frac{\text d u}{\text d t} = u_x \frac{\text d x}{\text d t} + u_t = f-u \, . $...


2

Let us introduce the vector ${\bf U} = (\theta, \eta)^\top$ of unknowns. Since $\partial_t {\bf U}$ is nonlinear in ${\bf U}$, the Crank-Nicolson method requires to invert the nonlinear algebraic system $F_\Delta (\theta,\eta) = 0$ from Eq. (30) at each time step. This step provides the updated values ${\bf U}_m^{n+1}$ from the current values ${\bf U}_m^{n}$....


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$$u(x,t)=f\left(t+\frac{x}{2}\right)+g\left(t-\frac{x}{2}\right)\qquad\text{ is OK.} \tag 1$$ Next you are supposed to determine the functions $f$ and $g$ according to the specified conditions. First condition : $$\quad u(x,t=0)=0=f\left(0+\frac{x}{2}\right)+g\left(0-\frac{x}{2}\right)$$ $$g\left(-\frac{x}{2}\right)=-f\left(\frac{x}{2}\right)$$ Let $X=-\...


1

It's due to continuity and connectedness. Let's write this a tiny bit cleaner and fill in the gaps explicitly. Suppose that there exists $x_0\in \Omega$, with $u(x_0)=M=\max_{\bar{\Omega}}u,$ and let $S=\{x\in \Omega: u(x)=M\}.$ We will show that this set is non-empty and both open and closed in $\Omega$. If we can show this, then connectedness will imply ...


1

Integrating by parts and using the divergence theorem, we can write \begin{align} (\operatorname{div}[(\mathbf{b}\cdot\nabla u)\mathbf{b}],v) &= \int_\Omega\operatorname{div}[(\mathbf{b}\cdot\nabla u)\mathbf{b}]v dx = \\ &= \int_\Omega\operatorname{div}[(\mathbf{b}\cdot\nabla u)\mathbf{b}v] dx - \int_\Omega(\mathbf{b}\...


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In an axisymmetric flow without swirl, $u^\theta = 0$. Thus from your formulas, $\omega^r = 0 = \omega^z$, and hence $\omega = \omega^\theta e_\theta$. Now apply the Biot-Savart law Wikipedia link $$ u = \nabla\times(-\Delta)^{-1} \omega.$$ This comes from the fact that $\omega = \nabla\times u$ and the double curl identity $$ \nabla\times(\nabla\times u) = \...


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