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Mathematics Qualifying Exam Question Bank

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Question

Math 6410 • ODE

Winter 2004 • Question 10

Question

Math 6410 • ODE Winter 2004 • Question 10

Conservation Laws. Find a solution to (5.) and prove that it is a weak solution. \[ \begin {cases} \dfrac {\partial u}{\partial t}+(1+u^2)\dfrac {\partial u}{\partial x}=0, & \text {for }x\in \mathbb {R}\text { and }t\ge 0,\\ u(x,0)= \begin {cases} 1, & \text {if }x<0,\\ 0, & \text {if }x\ge 0, \end {cases} & \text {for }x\in \mathbb {R}. \end {cases} \tag {5.} \]

Raw LaTeX

\textbf{Conservation Laws.}
Find a solution to (5.) and prove that it is a weak solution.
\[
\begin{cases}
  \dfrac{\partial u}{\partial t}+(1+u^2)\dfrac{\partial u}{\partial x}=0,
    & \text{for }x\in\R\text{ and }t\ge0,\\
  u(x,0)=
  \begin{cases}
    1, & \text{if }x<0,\\
    0, & \text{if }x\ge0,
  \end{cases}
    & \text{for }x\in\R.
\end{cases}
\tag{5.}
\]

Topic Tags

Distributions and Weak Solutions Poincare Maps and Periodic Solutions Scalar Conservation Laws Sobolev Spaces and Energy Estimates