Exam
6420-2024-05
Summer 2024 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
Show that \(u(x)=\frac {1}{4\pi |x|}\) is the fundamental solution of the Laplace equation in \(\mathbb {R}^3\), \(-\Delta u=\delta _{0}\) in the sense of distributions, i.e. \[ \int _{\mathbb {R}^3} u(x)\Delta \varphi (x)\,dx=-\varphi (0) \]
for all \(\varphi \) smooth and compactly supported. You should provide a fully detailed proof, not just a formal computation.
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Question 2
State (carefully) and prove the standard \(L^2\) Poincaré inequality in a smooth bounded domain \(U\).
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Question 3
Let \(b(x)\) a smooth vector field on a bounded domain \(U\) in \(\mathbb {R}^n\). Consider a smooth scalar solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+\nabla \cdot (b(x)u) & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]
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Show that if \(\nabla \cdot b = 0\) then for all \(t > 0\)
\[ \sup _{x\in U}u(x,t)\le \sup _U |f|. \] - Show that if \(\nabla \cdot b \ne 0\) then for all \(t > 0\) \[ \sup _{x\in U}u(x,t)\le e^{At}\sup _U |f| \quad \text {with } A=\sup _U \nabla \cdot b. \]
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Question 4
Let \(c,\mu >0\) Consider the wave equation with nonlinear fric- tional dissipation \begin{equation} \tag {1} \begin {cases} u_{tt}-c^2\Delta u+\mu u_t^3=f & \text {in } (0,T]\times \mathbb {R}^d,\\ (u,u_t)=(\phi ,\psi ) & \text {on } \{t=0\}\times \mathbb {R}^d. \end {cases} \end{equation}
Explain the notion of domain of dependence in the context of wave-type PDEs. Show finite speed of propagation for this PDE by showing, using an energy method, a natural upper bound on the domain of dependence of some point \((t_0,x_0)\). Hint: You do not need to concoct a complicated energy to get perfect energy conservation, just use a simple and standard energy where you can prove energy dissipation (i.e. monotone decreasing).
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Question 5
Let \(U\) be a bounded domain. Consider the energy
\[ I[v]:=\int _U \frac 12|\nabla v|^2\,dx \]on the admissible class
\[ \mathcal {A}=\left \{v\in C^2(U)\cap C^1(\overline {U}): v=0\text { on }\partial U,\ \int _U v\,dx=1\right \}. \]Suppose that there is \(u\in \mathcal {A}\) satisfying
\[ I[u]=\min _{v\in \mathcal {A}}I[v]. \]Show that \(u\) must solve the PDE boundary value problem \[ \begin {cases} -\Delta u=\lambda & \text {in } U,\\ u=0 & \text {on } \partial U. \end {cases} \]
Here \(\lambda \in \mathbb {R}\) is a Lagrange multiplier associated with the constraint. Find a formula for \(\lambda \) in terms of \(I[u]\).
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Question 6
Consider the scalar conservation law,
\[ u_t+\left (\frac 12 u^2\right )_x=0\quad \text {in } \mathbb {R}\times (0,\infty ). \]Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1-x, & 0<x<1. \end {cases} \]
Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.