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Fix a bounded smooth function $𝑔(𝑥)$. That is, both $𝑔(𝑥)$ and all derivatives of $𝑔(𝑥)$ are uniformly bounded on ${ℝ}^{𝑛}$. Fix also a constant $𝑣>0$. Consider the function

defined for $𝑡\ge 0$ and $𝑥\in {ℝ}^{𝑛}$.

(i) Show that $𝑢(𝑡,𝑥)$ is differentiable in $𝑡$ and $𝑥$ for almost every $(𝑡,𝑥)$.

(ii) Suppose that $𝑢(𝑡,𝑥)$ is continuously differentiable in a space-time neighborhood of a point $(𝑡,𝑥)$. Show that then

(iii) Give an example of a smooth function $𝑔(𝑥)$ such that $𝑢(𝑡,𝑥)$ is not differentiable in $𝑥$ everywhere for some $𝑡>0$.

(iv) Do there exist smooth functions $𝑔(𝑥)$ such that $𝑢(𝑡,𝑥)$ is differentiable in $𝑥$ everywhere for all $𝑡>0$?

Consider the PDE

with $𝑎(𝑥)\ge {𝑎}_0>0$ and $𝑐(𝑥)\ge {𝑐}_0>0$ smooth functions on $[0,1]$ and $𝑓(𝑥)\in {𝐿}^2$.

(i) Write down the bilinear form of this PDE in the ${𝐻}_0^1$ space on $(0,1)$.

(ii) Show that this bilinear form is bounded and coercive on ${𝐻}_0^1$.

(iii) Consider a piecewise linear finite element discretization with mesh size $ℎ$. Denote the finite approximation space by ${𝑆}_{ℎ}$ and the finite element solution by ${𝑢}_{ℎ}$. Show that there exist some positive constants ${𝐶}_1,{𝐶}_2$ such that

where ${\Vert \cdot \Vert }_1$ is the ${𝐻}_0^1$ norm.

Consider the equation

with initial condition $𝑝(𝑥,0)=𝛼(𝑥)$ and ${𝑝}_{𝑡}(𝑥,0)=𝛽(𝑥)$.

(i) Write down an equivalent symmetric hyperbolic system. Hint: use ${𝑝}_{𝑡}$ and ${𝑝}_{𝑥}$.

(ii) Write down the Friedrichs scheme for the system obtained in (i). Denote the time step by $𝑘$, the spatial step by $ℎ$, and the numerical approximation by ${𝑈}_{𝑗}^{𝑛}$.

(iii) Assume that the finite difference method is defined for all $𝑥\in ℝ$ and consider it as a map ${𝐸}_{𝑘,ℎ}$ from ${𝑈}^{𝑛}(𝑥)$ to ${𝑈}^{𝑛+1}(𝑥)=({𝐸}_{𝑘,ℎ}{𝑈}^{𝑛})(𝑥)$. Write down the Fourier multiplier ${\widetilde{𝐸}}_{𝑘,ℎ}$ using the convention:

(iv) Find the right condition on $ℎ$ and $𝑘$ so that the method is stable.

Consider

for a smooth convex flux function $𝑓(𝑢)$.

(i) Assume the domain $ℝ$ is partitioned into cells ${𝐶}_{𝑗}$ with size $\mathrm{\Delta }𝑥$ and center ${𝑥}_{𝑗}$. Describe the Godunov scheme. Discuss the constraint on the time step $\mathrm{\Delta }𝑡$.

(ii) Prove that the numerical flux of the Godunov scheme defined using the Riemann problems has the following form

Consider a Hamiltonian $𝐻(𝑞,𝑝)=\frac{1}{2}{𝑝}^2+𝑉(𝑞)$.

(i) Describe the Euler-B and Euler-A methods.

(ii) Prove that Euler-B and Euler-A both preserve the symplectic form.

(iii) Describe the Verlet method.

(iv) Explain why the Verlet method also preserves the symplectic form.

Suppose that $𝑢:{ℝ}^3\to ℝ$ satisfies $\mathrm{\Delta }𝑢\ge 6$.

(i) Show that

(ii) Show that there's ${𝑥}_{𝑖}\to \infty$ so that ${lim\; sup}_{𝑖\to \infty }{|{𝑥}_{𝑖}|}^{-2}𝑢({𝑥}_{𝑖})\ge 1$.

Suppose that $\mathrm{\Omega }\subset {ℝ}^{𝑛}$ is a compact domain with smooth boundary. Assume that $|\mathrm{\Omega }|=1$. Let $𝑢$ solve

where $𝜈$ is the outwards pointing unit normal to $𝜕\mathrm{\Omega }$. Assume that ${\int }_{\mathrm{\Omega }}{𝑢}_0=1$. Show that the time of existence satisfies $𝑇<1$.

Let $𝑓(𝑡)\in {𝐿}^{𝑝}(\mathrm{\Omega })$, where $\mathrm{\Omega }\subset {ℝ}^{𝑛}$. We say that

is the distribution function of $|𝑓|$.

(i) Show that

and

(ii) More generally, show that for any differentiable increasing function $𝜑(𝑠)$ such that $𝜑(0)=0$ we have

Let $𝜇$ be a positive measure on $𝑋$, $𝜇(𝑋)<\infty$, $𝑓\in {𝐿}^{\infty }(𝑋;d𝜇)$ and let

Prove that

Let

extend it periodically to all of $ℝ$, and define ${𝜑}_{𝑛}(𝑡)={𝜑}_0(2^{𝑛}𝑡)$, $𝑛\in \mathbb{N}$. Assume that $\sum {|{𝑐}_{𝑛}|}^2<\infty$ and show that the series

converges for almost every $𝑡$.