Questions tagged [transport-equation]
For questions related to transport equations. The transport equation describes how a scalar quantity is transported in a space.
157
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Asymptotic analysis of a linear advection-diffusion equation
Consider the linear advection-diffusion equation for $t,x>0$
\begin{equation}
\frac{\partial c}{\partial t} + f(x)\frac{\partial c}{\partial x} = \varepsilon \frac{ \partial^2 c}{\partial x^2} \tag{...
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vote
1
answer
85
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Solving the transport PDE equation
Consider the nonuniform transport equation
$$ u_t + (x^2 - 1) u_x = 0 $$
with initial condition $ u(0, x) = f(x)= e^{-x^2} $
I need to show that the solution is
$$
u(t, x) = \exp\left( -\left(\frac{x ...
3
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0
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Uniform Boundedness of a $C_0$-group
Let $p\in [1,\infty)$. On $X=L^p(\mathbb{R}; \mathbb{C}^2)$ we consider the operator $D=A+B$, where
$A=\begin{pmatrix}-\partial_x & 0 \\ 0 & \partial_x \end{pmatrix}\quad{and}\quad B=c(x)\...
7
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2
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ODE's: Continuity Equation
The context of this question is Machine Learning (more specifically, my question results from this paper, yet I have a math question, so I'm posting it here). First of all, some definitions (Sec. 2 of ...
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Finding the variational formulation of the following transport equation with boundary condition u'(1) + u(1) = 1
I'm studying variational problems and finite element method, and I'm trying to solve the following equation with boundary conditions:
for $u: ]0,1[ \to \mathbb{R}$,
$-((1 + x)u')'= x,$ for $x \in ]0,1[...
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0
answers
43
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transport equation interpretation
Premise: Theorem 5.34 from the book "Topics in Optimal Transportation"
Let $X$ be $\mathbb{R}^n$. Let $(T_t)_{0\leq t\leq T_{*}}$ be a locally Lipschitz family of diffeomorphisms in $X$, ...
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1
answer
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Decay estimates for transport equations in massless case
Considering the kinetic-transport equation $f(t,x,p)$ in the mass shell. It is well known that in the classic case (Vlasov non-relativistic)
\begin{align*}
\int_{\mathbb{R}^3} f(t,x,v)dv &= \...
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0
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Two-dimensional generalization of Leibniz's integral rule
Given a function $f(x,y):\mathbb R\times \mathbb R\to \mathbb C$ and a real parameter $\theta$, one can use Leibniz's integral rule to solve
\begin{equation}\label{eq}\tag{1}\frac{d}{d\theta}\int_{a(\...
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Realizing a modified transport equation
Stated somewhat informally, the continuity equation or transport equation $\partial_t\rho_t = -\nabla\cdot(\rho_t v_t)$ describes the evolution of a density where each particle flows along a vector ...
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Transport equation and entropy conditon
Consider the transport equation with smooth coefficient $a \in C^1(\mathbb{R}\times \mathbb{R}^+)$ given by
\begin{align}
u_t+(a(x,t)u)_x=0.
\end{align}
A weak solusolution $u \in C(\mathbb{R}^+;L^{1}(...
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1
answer
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Lie Bracket in curvilinear coordinates
When one works with Euler's equation in its vorticity formulation, one needs to work with the transport equation
$$(B\cdot \nabla)j=(j\cdot \nabla)B.$$
If one just uses cartesian coordinates this is (...
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1
answer
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What are the initial conditions for this coupled advection/transport system?
Consider the coupled transport (i.e. advection) system
$$
\begin{align}
\partial_t u + b\partial_x \phi &= 0,\\
\partial_t \phi + b\partial_x u &= 0,
\end{align}
$$
where $u(x,t),\phi(x,t) \in ...
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Understanding the construction to find the nonhomogeneous solution of a transport equation.
L. Evans book on PDE shows the solution for the nonhomogeneous transport equation in the following way:
\begin{cases}
u_t+b\cdot Du&=f & \text{in } \mathbb{R}^n\times(0,\infty)\\
u&=g &...
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Method of Characteristics vs given solution of Transport Equation
Definition of a transport equation:
$u_t+b\cdot u_x=0$ on $\mathbb{R}\times (0,\infty)$ and given the initial value problem:
\begin{cases}
u_t+tu_x=0, & (x,t)\in\mathbb{R}\times(0,\infty) \\...
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1
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Quasilinear Transport Equation
Assume $u: \mathbb{R^{2} \times \mathbb{R}_{\geq 0}} \to \mathbb{R}$ satisfy the following PDE
$$ \nabla u \cdot \langle 1, y, -x \rangle =u$$
where $\nabla u = \langle \partial_{t} u, \partial_{x} u,...
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2
answers
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What is the Steady-State of this Continuous-Space Discrete-Time Markov Chain when $k\leq 1/2$?
So, I asked a question earlier about an interesting Markov chain that I found, and I learnt that when $k\leq\frac12$, we start getting an ordinary distribution on a measureable set.
But I was ...
1
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1
answer
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Solution theory for Cauchy-Riemann Equations in the spirit of transport equation
This is just a peculiar thought I had while I was brushing up on my complex analysis notes and coincidentally having the (homog.) transport equation in mind:
Comparing the two set of equations
$$\...
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1
answer
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Solution to homogeneous Fredholm equation of the second kind with asymmetric kernel
I am trying to solve the homogeneous Fredholm equation of the second kind:
$w(r) = \frac{c}{2}\frac{1}{\frac{\gamma + a}{v \sigma b} + 1 - \frac{\gamma}{f\sigma} r} \int_{-1}^1 d r' w(r'),$ $\quad\...
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1
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Does transport equation preserve $L^\infty$ norm?
For transport equation with constant velocity $v$,
$$u_t+v\cdot \nabla u=0, u(x,0)=f(x)$$
By method of characteristics, we have $u(x,t)=f(x-vt)$. Thus
$$||u(x,t)||_{L^\infty}=||f||_{L^\infty}$$
i.e. ...
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1
answer
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Solve transport equation with variable velocity $v(x,t)$
I want to get the solution of
$$\frac{\partial u(x,t)}{\partial t}=v(x,t)\cdot \nabla u(x,t)$$
with $u(x,0)=f(x)$
The problem I encountered using method of characteristics is that: I cannot write the ...
3
votes
1
answer
1k
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Conservation law and entropy condition problem
Consider the scalar conservation quantity:
$u_t+(f(u))_x=0$ with $f(u)=u-u^2$, and initial condition
$u(x,0)=\begin{cases} c, x<0 \\ 1, 0<x<1 \\ 0, x>1\end{cases}$
I want to be able to ...
3
votes
1
answer
389
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Solving Quasi-Linear Transport Equation with two shockwaves.
I want to solve the following Partial Differential Equation by using the method of characteristics. This is the transport equation.
\begin{align}u_t - (1-2u)u_{x}&=0, &-\infty < x < \...
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
$$...
1
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0
answers
106
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Finite Difference Methods for 1D Transport Equation with Larger Stencil Widths
I'm attempting to re-create the numerical experiments in this paper for solving the 1D scalar advection (transport) equation$:$
$$
\frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0
$$
...
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votes
1
answer
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Advection reaction equation is solved by projection of solution of continuity equation
Suppose an absolutely continuous curve $\mu \colon (0, \infty) \to P_2(\Omega)$, where $P_2$ is the Wasserstein-2-space, fulfils the continuity equation
$$ \label{eq:CE} \tag{CE}
\partial_t \mu_t = \...
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1
answer
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Solve the first order nonhomogeneous PDE with I.C.
I am instructed to solve the following:
$$\frac{\partial U(x,t)}{\partial t}+c\frac{\partial U}{\partial x}=2(x-ct)$$
$$u(x,0)=f(x)$$
I said $\frac{dx}{dt}=C \implies x=ct+x_0$ and $\frac{dU}{dt}=g(t)=...
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1
answer
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Inhomogeneous linear transport equation Cauchy problem
I am working through the first chapter of "Finite Difference Schemes and Partial Differential Equations" by Strikwerda and I am confused by this inhomogeneous problem (1.1.2):
Show that
$$
...
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Initial data and a distributional Grönwall's inequality in the transport equation
I'm trying to understand the uniqueness result for renormalised solutions to the transport equation. A while back I tried to read the original source but could not get through it. Now I am trying to ...
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0
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Continuity equation for the density of a measure
From the paper of Ambrosio-Crippa, it is known that if $\beta:\mathbb R^d\times[0, T[\longrightarrow\mathbb R^d$ is suitably regular, then the system
$$
\begin{cases}
\frac{\partial\mu}{\partial t}(x, ...
3
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1
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Proper advection equation for non-conservative values in polar cylindrical coordinates.
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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2
answers
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weak solution of linear transport (advection) equation
How to show that $g(x-at)$ is the weak solution of the initial value problem
$$u_t+au_x=0$$
$$u(x,0)=g(x)$$ where $ g(x)\in L^{\infty}(\mathbb{R})$
Definition: $u$ is said to be the weak solution of ...
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1
answer
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Differentiation of $z(s)$
This is the book of Partial Differential Equation whose author is Lawrence C. Evans. I am reading the Linear Transport Partial Differential Equation of the First Order. In the formulation of the ...
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0
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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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1
answer
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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)=...
2
votes
2
answers
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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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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}(...
2
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0
answers
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Vorticity transport 2D using Lax-Wendroff 2 steps scheme
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$\zeta_t+u \cdot \zeta_x + v \...
1
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0
answers
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Heaviside as distributional solution for transport PDE?
I need to show that $H(x-ct)$ is a solution for the transport equation in the sense of distributions. I'm following the text Partial Differential Equations by Michael Shearer. Specifically, in Section ...
0
votes
1
answer
64
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Sufficient conditions for stability of numerical schemes for PDE?
I've got this numerical scheme for the 1D linear advection equation $u_t + au_x = 0$: $$\dfrac{U_{j, n+1} - U_{j, n}}{\Delta t} + a \dfrac{U_{j+2, n} - U_{j+1, n}}{\Delta x} = 0.$$ Tried von Neumann ...
3
votes
2
answers
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How do I properly implement a hyperbolic TVD flux limiter?
I'm trying to implement a TVD flux limiting scheme for the hyperbolic transport equation. However, I am currently stuck at implementing the limiter. How would I properly do this?
Here's how far I got:...
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1
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Advection Equation with initial data $u(x,0)=\sin(x)$
I am looking for analytic solution of the Advection Equation:
$$
\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).
$$
...
2
votes
1
answer
119
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Stabilizing Upwinding of Transport Equation with Varying Velocity and Large Gradients
I am attempting to solve the following transport equation using a 2D finite-difference scheme
$$c_t+\mathbf{v}(c)\cdot\nabla c=\frac{1}{Pe}\nabla^2 c,$$
where $c$ is modeling a local concentration, $...
3
votes
1
answer
84
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How to determine whether a solution of a transport PDE is classical?
I have to determine whether the solution of:$$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}{...
2
votes
2
answers
101
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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 ...
2
votes
2
answers
730
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Inhomogenous nonlinear transport equation $u_t+uu_x = -Du$
We have the following setup: $$u_t+uu_x = -Du \\ u(x,0)=\sin x.$$ The question is to find the time $T_s$ of a first shock formation. So basically, I need to solve the equation using method of ...
1
vote
0
answers
39
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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
$$\quad x_3 \geq -L,\quad p\in\...
1
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1
answer
84
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Problem to show the solution of the transport equation is a solution
Quick recall of the transport equation: I have the Cauchy problem
\begin{cases}
u_t(x,t)+ \langle c, \nabla_xu(x,t)\rangle=0, \text{ in }\mathbb{R}^n\times (0,a) \\ u(x,t)=g(x), \text{ on } \...
1
vote
0
answers
37
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Traffic Dynamics. How to fix initial condition deficiency?
Traffic in a tunnel. A rather realistic model for the car speed in a very long tunnel
is the following:
$$ v(\rho)=\left\{\begin{matrix}
v_m & 0 \leq \rho \leq \rho_c\\
\lambda log(\frac{\rho_m}{...
1
vote
1
answer
657
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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 ...
5
votes
2
answers
696
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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 ...