# Questions tagged [transport-equation]

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### Transport vs Continuity equation

For a time dependent vector field $v:\mathbb{R}^+\times \mathbb{R}^d\to\mathbb{R}^d$, and a say a (time dependent) probability density $u$, why do people call $$\partial_tu=\text{div }(uv),$$ the ...
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### Analytic solution of the following PDE

Consider the initial value problem: $u_t-(xu)_x = 0$, $\qquad$ $u(x,0) = \left\{\begin{array}{ll}\cos^2(\pi x/2), & -1\leq x \leq 1,\\0,&\text{otherwise}. \end{array}\right.$ I have tried ...
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### Prove that PDE $u_t+x^2u_x=0$ has infinite solutions

the text of the exercise I am takling is this one. Given \begin{cases} u_t+x^2u_x=0 \text{ in } (0, +\infty)\times \mathbf{R}\\ u(0,x)=0 \text{ in } \mathbf{R} \end{cases} prove that for every ...
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### How to use the Lie derivative to "perform" a parallel transport along a curve

Setup Consider a metric, for example that of a sphere with fixed radius $R$, i.e. $$ds^2 = R^2 d\theta^2 + R^2\sin^2\theta^2d\varphi^2,$$ and a curve on that sphere $\gamma = (\theta_0, \varphi)$, ...
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### Exact solution of the transport equation, Neumann boundary

Please, can anyone give me some examples of the exact solution of the following Pde with the corresponding functions f and g. I need these to know if my code of an approximation scheme is correct or ...
108 views

### Regularity of the weak solution of the transport equation

Consider the following transport equation: $$y_t(t,x)=y_x(t,x), \ \ (t,x)\in(0,\infty)\times(-\infty,\infty)$$ with initial state $y(0,x)=y_0(x)\in L^2(-\infty,\infty)$. By the method of ...
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If we consider the following transport equation with $t>0$ and $x\in \mathbb{R^3}$: $$\begin{cases} \partial_t f(t,x) + v(t,x). \nabla f(t,x)=0\\ f(0,x)=g(x) \end{cases}$$ And if we define the ...