Question

In each exercise, a Cauchy problem is given, with initial data specified on a curve $\gamma$. (a) Sketch the curve $\gamma$. (b) Determine the values of the parameter $\tau$, if any, where the transversality condition fails to hold. (c) Assume that $\omega (\tau )$ is continuously differentiable on the given interval $\alpha \le \tau \le \beta .$ Are all the hypotheses of Theorem $10.1$ satisfied? If not, which hypotheses do not hold? $2u_x+u_t=0,u(\tau ,\tau /2)=\omega (\tau ),0\le \tau \le 10$

Short answer

Answer: If all the hypotheses of Theorem 10.1 are satisfied, we can conclude that there exists a unique solution to the Cauchy problem in the domain of interest.

Step-by-step solution

Sketch the curve $\gamma$

For the given problem, we have the curve $\gamma$ defined by $u(\tau ,\tau /2)=\omega (\tau )$ for $0\le \tau \le 10$. The curve is a straight line in the $(x,t)$ plane, with slope ${\displaystyle \frac{1}{2}}$. So we have a straight line going from $(0,0)$ to $(10,5)$. Hence, $\gamma$ is the line connecting these two points.

Determine the values of $\tau$ where the transversality condition fails

To find if the transversality condition fails for any value of $\tau$, we need to check if the normal vector to the curve $(\frac{d\tau }{dt},\frac{d\tau /2}{dt})$ is parallel to the characteristic speeds $(2,1)$ for any $\tau$. We have $\frac{d\tau }{dt}=1$ and $\frac{d\tau /2}{dt}=1/2$. Hence, the tangent vector to the curve is $(1,1/2)$. The condition for the normal vector to be parallel to the characteristic speeds is if they are scalar multiples of each other. However, no scalar multiple of $(1,1/2)$ is equal to $(2,1)$. Thus, the transversality condition does not fail for any value of $\tau$, $0\le \tau \le 10$.

Check if the hypotheses of Theorem 10.1 are satisfied

Theorem 10.1 has the following hypotheses: (a) The initial function $\omega (\tau )$ is continuously differentiable on the interval $[0,10]$. (b) The given PDE has smooth coefficients. (c) The transversality condition holds. Here, we have not been given any information about the function $\omega (\tau )$, so we cannot directly check hypothesis (a). Nevertheless, we can assume that $\omega (\tau )$ is continuously differentiable on the given interval, as stated in the problem. The given PDE has constant coefficients, which are smooth functions. Hence, hypothesis (b) is satisfied. As we found in step 2, the transversality condition does not fail for any value of $\tau$, so hypothesis (c) is satisfied. Thus, if we assume the given condition about $\omega (\tau )$, all the hypotheses of Theorem 10.1 are satisfied.