# HW6 (MTH847 Fall 2017)

### General Info

Remember: for every question you must indicate all the people with whom you collaborated. If you didn't collaborate with anyone else on a given problem, write "Collaborators: none."

This problem set will be due in class, Wednesday November 29.

### Reminder of terminology:

• Let $$\Omega\subset\mathbb{R}^d$$ be an open domain, $$X\in C^0(\Omega,\mathbb{R}^d)$$ a continuous vector field on $$\Omega$$, and $$\Sigma\subset \Omega$$ some subset. We say that "the Cauchy problem for $$X\cdot \nabla u = 0$$ is solvable" if for every $$f\in C^1(\Omega,\mathbb{R})$$ there exists a solution $$u\in C^1(\Omega,\mathbb{R})$$ satisfying $$u |_{\Sigma} = f$$.
• A $$C^1$$ hypersurface $$\Sigma$$ is said to be non-characteristic with respect to the vector field $$X$$ if $$X$$ is never tangent to $$\Sigma$$.
• A $$C^1$$ hypersurface $$\Sigma$$ is said to be a Cauchy hypersurface with respect to the domain $$\Omega$$ and the vector field $$X$$ if $$\Sigma$$ is non-characteristic with respect to $$X$$, and that every maximally extended integral curve of $$X$$ on $$\Omega$$ intersects $$\Sigma$$ exactly once.

### Problems

1. Prove that when $$\Omega = \mathbb{R}^2$$, $$X = \partial_x$$, and $$\Sigma = { x = |y|}$$, the corresponding Cauchy problem is not solvable. (That is to say, produce a $$C^1$$ function $$f$$ on $$\mathbb{R}^2$$ such that if $$u$$ is any other $$C^1$$ function on $$\mathbb{R}^2$$ that agrees with $$f$$ on $$\Sigma$$, then $$\partial_x u$$ cannot vanish identically.)
2. Let $$\Omega = \mathbb{R}^2$$ and take $$X = \partial_x + \sin(x) \partial_y$$.
• Find the integral curves of $$X$$ and justify that the set $$\Sigma = {0}\times\mathbb{R}$$ is a Cauchy hypersurface for this $$\Omega$$ and $$X$$.
• Find an explicit solution formula for the Cauchy problem $$X \cdot \nabla u = 0$$ with $$u(0,y) = f(y)$$.
• Find an explicit solution formula for the inhomogeneous Cauchy problem $$X \cdot \nabla u = g$$ with $$u(0,y) = 0$$.
3. Let $$\Omega = \mathbb{R}^2$$ and take $$X = (1 + x^2) \partial_x + \partial_y$$. Let $$\Sigma = { y = 0}$$.
• Prove that $$\Sigma$$ is non-characteristic, but not a Cauchy hypersurface.
• Prove that the initial value problem for $$X\cdot \nabla u = 0$$ with $$u(x,0) = f(x)$$ has infinitely many solutions in $$C^1(\mathbb{R}^2)$$ when $$f(x) \in C^1_c(\mathbb{R})$$.
• Find initial data $$f(x) \in C^1(\mathbb{R})$$ such that the initial value problem above has no solutions in $$C^1(\mathbb{R}^2)$$.
4. Let $$\Omega = \mathbb{R}^2$$ and take $$X = (y-1) \partial_y$$.
• Prove that the hypersurface $$\Sigma = { y = 0}$$ is not a Cauchy hypersurface.
• Prove that nonetheless, given any initial data $$f\in C^1(\mathbb{R})$$, there exists a unique solution $$u\in C^1(\mathbb{R}^2)$$ to $$X \cdot \nabla u = 0$$ satisfying $$u(x,0) = f(x)$$.
(Remark: this means that the Cauchy hypersurface condition is sufficient but not necessary for solvability.)
5. Let $$\Omega, X, \Sigma$$ be as in question 2. Consider the nonlinear transport equation $X \cdot \nabla u = u^2, \qquad u(0,y) = f(y).$ Show that for any $$f\in C^1(\mathbb{R})$$ that is not identically zero, there does not exist a solution $$u \in C^1(\Omega)$$.
(Hint: let $$\gamma$$ be an integral curve of $$X$$, then along the curve $$u$$ must solve the ordinary differential equation $$\frac{d}{dt} (u \circ \gamma) = (u\circ \gamma)^2$$.)
6. Let $$P(z)$$ be a non-constant polynomial. Consider the quaslilinear transport equation $\partial_t u + P(u) \partial_x u = 0$ with initial data $$u(0,x) = f(x)$$. Prove that if $$f(x)$$ is not constant, there cannot exist a solution $$u\in C^1(\mathbb{R}\times\mathbb{R})$$.
7. Suppose $$u\in C^1(\mathbb{R}\times\mathbb{R}^d)$$ is a solution to the Hamilton-Jacobi equation $\partial_t u + \frac12 |\nabla u|^2 = 0$ where the symbol $$\nabla$$ denotes the gradient with respect to the spatial variables. Prove that, if $$x \mapsto u(0,x)$$ has compact support, then $$u \equiv 0$$.
8. Consider the initial value problem for the fully nonlinear first order PDE $(\partial_t u)^2 + \sin(\partial_t u) \partial_x u = 0$ for $$u = u(t,x)$$, with initial data $u(0,x) = c x$ where $$c$$ is a non-zero real constant. Prove that a (local-in-time) solution exists, that the solution is not unique, and that there are only finitely many solutions.
9. Consider the Cauchy problem for the fully nonlinear first order PDE $(\partial_x u)^2 - \frac{x^2}{1+x^2} e^{-2y} (\partial_y u)^2 = 0$ with initial data prescribed to vanish on the set $\Sigma := { \sqrt{1 + x^2} + e^y = 2 }.$
• Write down the characteristic system for this equation.
• Show that the solutions must be constant along integral curves of the characteristic vector field.
10. (Extra credit) Show that the Cauchy problem described in the previous question has infinitely many solutions.
(Hint: by using the initial data and the nonlinear PDE itself, one can conclude that the "direction" of the characteristic vector field is "independent of the solution". This means that up-to-reparametrization the integral curves of the characteristic vector field is independent of the solution $$u$$.)