Problem set 1

1. Use the method of characteristics to solve \[ \frac{\pd u}{\pd t} + x\frac{\pd u}{\pd x} = u, \qquad u(x, 0) = \frac{1}{1+x^2}. \] for $(x,t) \in \R\times [0, \infty)$.
2. Consider the linear first order PDE \[ \frac{\pd u}{\pd t} + x^2\frac{\pd u}{\pd x} = 0, \qquad u(x, 0) = \varphi(x) \] for $(x,t) \in \R\times [0, \infty)$, where $\varphi:\R\to\R$ is an odd real analytic function with $\varphi'(x)\gt0$ for all $x\in\R$, and for which $\lim_{x\to\pm\infty} \varphi(x) = \pm A$ exists (e.g. $\varphi(x)= \arctan x$, or $\varphi(x)=\tanh x$, or $\varphi(x) = x/\sqrt{1+x^2}$).
   1. Use the method of characteristics to compute $u$.
   2. Is the solution a real analytic function?
   3. Compute $\lim_{t\to\infty} u(x, t)$ for each $x\in\R$.
3. Let $F:\R\to\R$ be continuously differentiable. Assume that $u_n\in L^\infty(U)$ and $g_n\in L^\infty_(\R)$ are two sequences of functions such that $u_n$ is an integral solution of \[ \frac{\pd u_n}{\pd t} + \frac{\pd F(u_n)}{\pd x} = 0, \qquad u_n(x, 0) = g_n(x) \] for every $n\in\N$. Assume that \[ \lim_{n\to\infty} u_n(x, t) = u(x, t), \qquad \lim_{n\to\infty} g_n(x) = g(x) \] exist for almost every $(x, t)\in\R^2$ (for the $u_n$) and almost every $x\in\R$ (for the $g_n$). Assume also that there is an $M\gt0$ such that \[ |u_n(x, t)|\leq M, \qquad |g_n(x)|\leq M \] for all $n\in\N$ and almost all $(x, t)\in\R^2$ or almost all $x\in\R$.
   Prove that $u(x, t)$ is an integral solution of the conservation law with initial values given by $g$.