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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Exercise on transport equation

I have to solve the following exercise: \begin{equation} \begin{cases} \partial_t u(t,x)-7\partial_x u(t,x)=0\\ u(t=0,x)=e^{-x^6} \end{cases} \end{equation} And I found the classical solution $u(t,x)=...
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Well-posedness of vector-valued linear transport equation on time-varying domains

I have a vector-valued transport equation on a smoothly time-varying domain $\Omega = \Omega (t) \subset \mathbb{R}^2$ for the variable $u(x,t)$ assuming values in $\mathbb{R}^2$: $\dfrac{\partial u}{\...
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Existence of measure-preserving Lagrange flow for inhomogeneous transport equation

Let us consider the Cauchy problem for the transport equation $$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,\cdot)=\varphi_0, $$ where $\text{div}(...
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Heaviside as distributional solution for transport PDE?

I need to show that $H(x-ct)$ is a solution for the transport equation in the sense of distributions. I'm following the text Partial Differential Equations by Michael Shearer. Specifically, in Section ...
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Advection Equation with initial data $u(x,0)=\sin(x)$

I am looking for analytic solution of the Advection Equation: $$ \frac{\partial{u(x,t)}}{\partial{t}} + c \frac{\partial{u(x,t)}}{\partial{x}}=0, $$ with initial condition $$ u_0(x)=u(x,0)=\sin(x). $$ ...
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How to determine whether a solution of a transport PDE is classical?

I have to determine whether the solution of:$$u_y+2u_x=0 , x>0,y>0 ,\quad u(x,0)=x ,x\geq 0 ,\quad u(0,y)=y ,y\geq0$$ is a classical solution. I found that the solution is:$$u(x,y)=\frac{-x+2y}{...
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2 votes
2 answers
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Making Sense of Method of Characteristics Solution Geometry

How can I make sense of the following surface as insight into understanding the method of characteristics? The partial differential equation this initially came from was $2xu_x + u_y = 0$, which has ...
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Probabilistic Representation of Transport Equations

Consider the closed system of transport equations $$ \frac{\partial G_p}{\partial x_3} + \frac{2p}{\bar c}\frac{\partial G_p}{\partial\tau} = (\mathcal{L}G)_p,$$ where $$\quad x_3 \geq -L,\quad p\in\...
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Problem to show the solution of the transport equation is a solution

Quick recall of the transport equation: I have the Cauchy problem \begin{cases} u_t(x,t)+ \langle c, \nabla_xu(x,t)\rangle=0, \text{ in }\mathbb{R}^n\times (0,a) \\ u(x,t)=g(x), \text{ on } \...
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Traffic Dynamics. How to fix initial condition deficiency?

Traffic in a tunnel. A rather realistic model for the car speed in a very long tunnel is the following: $$ v(\rho)=\left\{\begin{matrix} v_m & 0 \leq \rho \leq \rho_c\\ \lambda log(\frac{\rho_m}{...
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Green's function of non-homogeneous advection equation

I am working on a physics problem which leads to an inhomogeneous PDE $$ \left[ \frac{\partial}{\partial t} + \vec{v} \cdot \nabla_\vec{r} \right] f(\vec{r},\vec{p},t) = g(\vec{r},\vec{p},t) $$ which ...
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How do the characteristics of a transport equation determine where we need to impose boundary conditions?

Consider the one-dimensional linear transport equation $$u_t+u_x=0\;\;\;\text{in }(0,\infty)\times\Omega;\tag1$$ $$u(0,\;\cdot\;)=u_0\;\;\;\text{in }\Omega.\tag2$$ If $\Omega=\mathbb R$, these ...
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Is the weak solution of the transport equation $\left(\frac\partial{\partial t}+b\cdot\nabla_x\right)u=0$, $u(0)=u_0$, still be given by $u_0(x-tb)$?

Let $d\in\mathbb N$, $b\in\mathbb R^d$ and $u_0:\mathbb R^d\to\mathbb R$ be differentiable. We can easily show that the unique (classical) solution of \begin{align}\left(\frac\partial{\partial t}+b\...
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TVD Lax-Wendroff with non-constant velocity

I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D. The equation is the following: \begin{equation} \frac{\...
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1 answer
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Prove a given function solves the transport equation PDE

From Evans' PDE, chapter 2.1.2 Prove $u(x,t)=g(x-tb)+\int_0^t f(x+(s-t)b,s)\,ds, u\in C^1(\mathbb R^n\times[0,\infty))$ solves \begin{cases} u_t+b\cdot Du=f &\text{in }\mathbb R^n\times(0,\infty),...
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Fully-implicit scheme for reactive transport equation.

I am reading a paper by Zhang 2007 about reactive chemical transport where a fully implicit method was used in solving the transport equation given by, "At $\left(n+1\right)$th time step, ...
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Later shocks in Riemann problem [duplicate]

I am following section 3.4 of Evan's book "Partial Differential Equations". In this section he considers the Burgers equation, $$u_t + \bigl(\frac{u^2}{2}\bigr)_x = 0$$ with initial data $$g(...
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Homogeneous transport PDE with data along $t=0$ and $x=0$

I am finding an analytic expression for the solution of the transport PDE: $$u_t+\left(\frac{1-2u(x,t)}{a}\right)u_x = 0,\quad a= \text{const.}, \quad x>0, \quad t >0$$ $$u(x=0,t) = u_0, \quad ...
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Energy estimate for a friedrich system with relaxation and periodic boundary conditions

I'm considering the following system of pdes $$ \partial_t u + \sum\limits_{i=1}^3 \mathcal{A}_i\partial_{i} u = -\mathcal{R} u. $$ Where the matrices $\mathcal{A}_i$ are symmetrics, $u$ is valued in ...
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Solving $u_t = (f(t)-1)u_x$

Good afternoon, Here's the question I'm currently working on: Determine the solution of the following transport equation: $$u_t = (f(t)-1)u_x \space \space \space ,\space x \in \mathbb{R}, \space 0 &...
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3 votes
1 answer
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Proper advection equation for non-conservative values in polar cylidrical coords.

I'm numerically solving a system of PDE's consiting of some conservation laws (Euler equations) along with advection of non-additive values (like molar mass or unit heat capacity, for example). The ...
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Adaptedness of solution to transport equation with random coefficients

While seemingly "obviously true", I fail to verify the following question. Let $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \leq T}, P)$ be a complete probability space with normal filtration. ...
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Simple Transport $u_t+cu_x=0$, why is the substance transported to the right at a fixed speed c?

In Partial Differential Equations by Walter Strauss, Ch 1.3, Example 1, they present the transport equation $u_t+cu_x=0$ whose solution is $u(x,t)=f(x-ct)$. They claim that this solution means the ...
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1 answer
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Literature on the PDE $\partial_t f + \partial_x(f(1-f))=0$ [closed]

I am looking for literature on this variant of the transport PDE: $$\partial_t f + \partial_x(f(1-f))=0.$$ Do you have any suggestions?
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1 answer
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How do we know the uniqueness of the transport equation?

I'm learning PDE and have a basic question. In Evans book on PDE, in order to solve $\partial_t u+\langle b, Du\rangle=0$ with the initial condition $u(0,\cdot)=g$, I first have to use the auxillary ...
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Regular Lagrangian flows on a domain of $\mathbb{R}^d$ with a boundary

I'm looking for some references about the theory of regular Lagrangian flows on a smooth domain $\Omega$ of $\mathbb{R}^d$ (say a smooth bounded open set of $\mathbb{R}^d$ or a half space). Here, ...
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2 answers
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$u_t+(u(1-u))_x=a(1-2u)$, transients to steady solution

We consider the non conserving equation $$u_t+(f(u))_x=af'(u)$$ where $a$ is a constant, 0$\leq$ x $\leq$ 1 and $f(u)=u(1-u)$. The steady solution of this equation with boundary condition $u(0)=u_0$ ...
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2 votes
2 answers
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$u_t+(u(1-u))_x=a(1-2u)$, method of characteristics for traffic flow equation with riemann initial data

We consider the non conserving equation $$u_t+(f(u))_x=af'(u)$$ where $a$ is a constant and $f(u)=u(1-u)$. I am trying to solve this equation by method of characteristics with the initial condition $$...
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Sufficient conditions for stability of numerical schemes for PDE?

I've got this numerical scheme for the 1D linear advection equation $u_t + au_x = 0$: $$\dfrac{U_{j, n+1} - U_{j, n}}{\Delta t} + a \dfrac{U_{j+2, n} - U_{j+1, n}}{\Delta x} = 0.$$ Tried von Neumann ...
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Cauchy Problem For Inhomogeneous Transport Equation

Here is a problem: $\left\{\begin{array}{l} 2 u_{t}-u_{x}=f(t, x), \quad t>0 \\ \left.u\right|_{t=0}=\varphi(x) \end{array}\right.$ My solution $\frac{d t}{2}=\frac{d x}{-1}=\frac{d y}{f(x, c)}$ ...
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What are the solutions of this differential operator?

I am thinking about the ‘complex’ transport equation: $$\partial_t + a\cdot \nabla = 0$$ where $a \in \mathbb{C}^n$ and its elements are either real or pure imaginary. The simplest example for this ...
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uncoupled transport equation of (U) and another coupled linear pde, does $U$ decay?

Let $C = 1$ and $\gamma = 1$. and the initial conditions as $U(t=0) = e^{cos(2\pi x)}$ (periodic boundary) and $V(t=0) = 0$. Now I'm getting two numerical solutions. One models the $U$ as a ...
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The Words 'Transport Equation'

Hi I come from a mathematical stochastic background. In PDE / ODE / analysis I sometimes hear the terminology 'transport equation' or 'transport part of the equation'. Can anyone explain $\textbf{In ...
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Characteristics of the Milne Problem on a Finite Slab

I'm trying to justify equation (3.43) on page 18 of this paper by Lei Wu and Yan Guo on diffusion approximation of the radiative transport equation. Consider the following penalized transport equation:...
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Parallel transport on the Stiefel manifold

I am reading the paper Alan Edelman, Tomas A. Arias, Steven T. Smith, The geometry of algorithms with orthogonality constraints, SIAM Journal on Matrix Analysis and Applications, Volume 20, Number 2, ...
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Steady state of diffusion-advection on the torus

Let $P$ be a positive scalar function and $\mathbf{v}(\mathbf{x})$ is an assigned smooth vector field. The quantity $P(t,\mathbf{x})$ evolves according to a transport equation of the kind $$ \...
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2 votes
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Average velocity of overdamped particles in external field

In short: how to obtain the average velocity from the Fokker-Planck equation in the overdamped regime? (i.e. when the probability density is $P(\mathbf{x},t)$ and not $P(\mathbf{x},\mathbf{v},t)$, ...
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How do I solve the following initial value problem using the method of characteristics?

I'm starting to learn about the method of characteristics and tried the problem below, but ran into some problems: $\begin{align*} v_t + \sin^2(x)v_x &= 0, \ t > 0, \ x \in(-\pi, \pi)\\ v(x,0) ...
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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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9 votes
2 answers
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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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Inhomogeneous transport equation with terminal value available

From this video I understand that in case of the inhomogeneous transport equation $$ u_t + c u_x = g (t,x)\tag{$\ast$}$$ with initial value $u (0,x) = \tilde{h} (x)$, the solution may be written as $$...
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Entropy condition for linear transport equation

Consider the initial value problem, \begin{eqnarray} u_t+f(u)_x=0 &\quad (x,t) \in \mathbb{R} \times \mathbb{R^+}\\ u(x,0)=u_0(x) &\quad x\in \mathbb{R} \end{eqnarray} For $f$ Lipschitz and $...
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2 answers
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Solving the "Transport" PDE in the sense of distributions with Dirac Delta Source

Let $\delta_0$ be the standard Dirac Delta distribution. I wish to solve the PDE $$u_t+cu_x=\delta_0$$ in the sense of distributions with initial condition $u(x,0)=g(x)$ for some continuous $g$. That ...
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Show that the initial value problem for the transport problem defines a contraction semigroup

I'm working through some problems with contraction semigroup right now. Show that the initial value problem for the transport equation: $$ \frac{\partial u}{\partial t} + c \frac{\partial u}{\...
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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 ...
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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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Source for Theory of Linear Transport equations

I'm looking for a source, like a textbook or something else that be used as a reference in a paper, for the theory of linear transport equations. I want to have a proof of the well-known result that ...
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3D linear transport equation solution

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 ...
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