Topic
Scalar Conservation Laws
Courses
Courses whose questions include this topic tag.
Questions
This topic is tagged for 6 questions
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Fall 2025 • Question 4
Find the entropy solution of the following scalar conservation law, arising in traffic flow modeling,
\[ \begin {cases} \partial _t u+\partial _x(u(1-u))=0 & \text {for } x\in \mathbb {R},\ t>0,\\ u(x,0)=g(x) \end {cases} \]with initial data
\[ g(x)= \begin {cases} 0, & x<-1,\\ 1, & -1<x<0,\\ 0, & 0<x<1,\\ \frac 12, & 1<x. \end {cases} \]After the rarefaction hits the shock you should write the explicit ODE IVP for the shock location \(\gamma (t)\) and a piecewise definition of the solution using this shock location \(\gamma (t)\), but you don’t actually need to solve the ODE IVP to get the explicit formula for \(\gamma (t)\).
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Fall 2024 • Question 4
Consider the scalar conservation law,
\[ u_t+\left (\frac 14 u^4\right )_x=0\quad \text {in } \mathbb {R}\times (0,\infty ). \]Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \]
Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.
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Summer 2024 • Question 6
Consider the scalar conservation law,
\[ u_t+\left (\frac 12 u^2\right )_x=0\quad \text {in } \mathbb {R}\times (0,\infty ). \]Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1-x, & 0<x<1. \end {cases} \]
Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.
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Fall 2004 • Question 10
Conservation Laws. A model for flow of fluid in an infinite pipe says that the density \(\rho (x,t)\) satisfies \[ \frac {\partial \rho }{\partial t}+\rho \frac {\partial \rho }{\partial x}=-\rho ^2,\qquad \rho (x,0)= \begin {cases} 1, & \text {if }x\le 0,\\ \frac {1}{2}, & \text {if }x>0. \end {cases} \]
- Find a solution \(\rho \).
- Define what it means for your solution \(\rho \) from part (a) to be a weak solution of the system. (Do not carry out the calculations to prove that \(\rho \) is a weak solution.)
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Winter 2004 • Question 10
Conservation Laws. Find a solution to (5.) and prove that it is a weak solution. \[ \begin {cases} \dfrac {\partial u}{\partial t}+(1+u^2)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)= \begin {cases} 1, & \text {if }x<0,\\ 0, & \text {if }x\ge 0, \end {cases} & \text {for }x\in \mathbb {R}. \end {cases} \tag {5.} \]
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Fall 2003 • Question 7
Conservation Laws. Consider the first order equation \[ \begin {cases} \dfrac {\partial u}{\partial t}+(\tan u)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)=\varphi (x), & \text {for }x\in \mathbb {R}. \end {cases} \qquad (\diamond ) \]
- Find a solution \(u(x,t)\) to \((\diamond )\) where \(\varphi (x)=\operatorname {Arctan}(x)\).
- For \[ \varphi (x)= \begin {cases} 0, & \text {if }x<0,\\ \frac {\pi }{4}, & \text {if }x>0, \end {cases} \] find two weak solutions to \((\diamond )\). Prove that one of them is a weak solution.
- State the entropy (Rankine-Hugoniot) condition. Which weak solution of part (b.) satisfies this condition?