### 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."

Please write legibly, or type-up your solutions in LaTeX. The grader will greatly appreciate it.

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

- 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.) - 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 \(u(0,y) = 0\).

- 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 \(C^1(\mathbb{R}^2)\).

- 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.*)

- 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 \(\frac{d}{dt} (u \circ \gamma) = (u\circ \gamma)^2\).*) - 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 \( u\in C^1(\mathbb{R}\times\mathbb{R})\).
- 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 \(u \equiv 0\).
- 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. - 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.

- (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 \(u\).*)