Skip to main content

Questions tagged [transport-equation]

For questions related to transport equations. The transport equation describes how a scalar quantity is transported in a space.

Filter by
Sorted by
Tagged with
1 vote
0 answers
47 views

semilinear transport equation

Good morning everyone! I have to solve the following transport equation for the temperature $T(x,t)$: $$ \frac{\partial T}{\partial t} + v \frac{\partial T}{\partial x} = k(T_a - T) $$ with boundary ...
Gregorio Coletti's user avatar
0 votes
0 answers
38 views

Transport equation with dirichlet boundaries counditions

my fellows mathematicians, I have a question concerning the linear transport equation in a very simple case (1D with Dirichlet boundaries conditions) \begin{equation} \begin{split} & f : x \in ...
leo's user avatar
  • 1
0 votes
0 answers
35 views

Writing a PDE in a moving coordinate

I am trying to understand the following PDE by writing it in a reference frame marked by the red cross. $\partial_t h(x,t)+ \partial_x (u(x,t) h(x,t))=0,$ where h shows the height. I want to write an ...
questionerno8's user avatar
0 votes
0 answers
16 views

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{...
Giraffes4thewin's user avatar
1 vote
1 answer
106 views

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 ...
Tomer's user avatar
  • 436
3 votes
0 answers
88 views

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)\...
ym94's user avatar
  • 873
0 votes
0 answers
45 views

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[...
Amsel's user avatar
  • 11
1 vote
0 answers
51 views

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 $X$, ...
chintan's user avatar
  • 39
0 votes
1 answer
100 views

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 &= \...
Gaiüx's user avatar
  • 3
1 vote
0 answers
165 views

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(\...
A Quantum Field Day's user avatar
1 vote
0 answers
21 views

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 ...
Juno Kim's user avatar
  • 610
4 votes
0 answers
87 views

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}(...
Rosy's user avatar
  • 1,025
1 vote
1 answer
68 views

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 (...
DaniS's user avatar
  • 13
0 votes
1 answer
56 views

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 ...
l'étudiant's user avatar
0 votes
0 answers
24 views

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 &...
JackpotWizard 180's user avatar
0 votes
0 answers
40 views

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) \\...
JackpotWizard 180's user avatar
0 votes
1 answer
43 views

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,...
Matha Mota's user avatar
  • 3,492
2 votes
2 answers
102 views

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 ...
Thomas Pluck's user avatar
7 votes
2 answers
594 views

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 ...
Hermi's user avatar
  • 702
1 vote
1 answer
56 views

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 $$\...
Taleofwoe's user avatar
  • 109
0 votes
1 answer
188 views

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. ...
Wang's user avatar
  • 105
0 votes
1 answer
90 views

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 ...
Wang's user avatar
  • 105
1 vote
0 answers
121 views

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$:$ $$ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x} = 0 $$ ...
voyseto's user avatar
  • 11
0 votes
1 answer
54 views

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)=...
Christine's user avatar
0 votes
1 answer
75 views

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\...
user1290's user avatar
5 votes
1 answer
130 views

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 = \...
ViktorStein's user avatar
  • 4,888
1 vote
1 answer
132 views

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 $$ ...
Eilif's user avatar
  • 411
1 vote
0 answers
89 views

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, ...
Redeldio's user avatar
  • 551
6 votes
0 answers
133 views

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 ...
Calvin Khor's user avatar
  • 35.2k
0 votes
1 answer
41 views

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 ...
Nirmal Rawat's user avatar
1 vote
0 answers
220 views

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 ...
Terminology Terry's user avatar
1 vote
1 answer
98 views

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)=...
Pefok's user avatar
  • 664
1 vote
0 answers
62 views

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}(...
user99432's user avatar
  • 890
1 vote
0 answers
174 views

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 ...
1Teaches2Learn's user avatar
0 votes
1 answer
221 views

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 $$ u_0(x)=u(x,0)=\sin(x). $$ ...
user148733's user avatar
3 votes
1 answer
86 views

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}{...
george_ioanidis's user avatar
2 votes
2 answers
102 views

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 ...
user10478's user avatar
  • 1,922
1 vote
0 answers
41 views

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\...
rami_salazar's user avatar
1 vote
1 answer
85 views

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 } \...
Drokox's user avatar
  • 13
3 votes
0 answers
59 views

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}{...
DANNA KATHERINE NIETO HUERTAS's user avatar
1 vote
1 answer
745 views

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 ...
Pascal S.'s user avatar
  • 343
0 votes
0 answers
246 views

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\...
0xbadf00d's user avatar
  • 13.9k
1 vote
0 answers
184 views

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{\...
Iddingsite's user avatar
2 votes
1 answer
182 views

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),...
sound wave's user avatar
0 votes
0 answers
41 views

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, ...
FlickerBeat's user avatar
1 vote
0 answers
92 views

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(...
CBBAM's user avatar
  • 6,275
1 vote
1 answer
71 views

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 ...
Alando's user avatar
  • 25
1 vote
1 answer
46 views

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 ...
Djekt's user avatar
  • 129
0 votes
1 answer
72 views

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 &...
Daniel F.'s user avatar
3 votes
1 answer
648 views

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 ...
omican's user avatar
  • 33