querent

(i) Find a unique pair of numbers ${𝛼}_{𝑛}>0$ and ${𝛽}_{𝑛}>0$ such that an inequality of the form

holds for all $𝑢\in \mathrm{𝒮︀}({ℝ}^{𝑛})$.

(ii) Show that this inequality does hold.

(iii) Use that inequality to show that a smooth rapidly decaying solution to a parabolic equation of the form

satisfies an estimate of the form

with a constant $𝐶[𝑎]$ that depends only on the ellipticity constant $\mathrm{\Lambda }$ of the matrix $𝑎(𝑥)$. In other words, for all $𝑥\in {ℝ}^{𝑛}$ and $𝜉\in {ℝ}^{𝑛}$ we have

Consider the solutions to the viscous Burgers' equation

in dimension one, with some $𝜈>0$.

(i) Show that if $𝑢(0,𝑥)={𝑢}_0(𝑥)$ is continuously differentiable, then $𝑢(𝑡,𝑥)$ is infinitely differentiable for all $𝑡>0$.

(ii) Show that for all $𝑡>0$, $𝑥\in ℝ$ and $𝑦>0$ we have

Show that there is a constant $𝐶>0$ so that for any smooth function $𝑢(𝑥)$ such that $𝑢(𝑥)\to 1$ as $𝑥\to -\infty$ and $𝑢(𝑥)\to 0$ as $𝑥\to +\infty$, we have

Find the Euler-Lagrange equations for the functional

with two different functionals

and

Show that if a distribution $𝜑\in {\mathrm{𝒟︀}}^{\prime }(ℝ)$ satisfies ${𝜑}^{\prime }=0$ (in the sense of distributions), then $𝜑=𝑐$, where $𝑐\in ℝ$ is a constant.

(i) Let $𝑢(𝑡,𝑥)$ be a smooth solution to a conservation law

Assume that $𝑓\in {𝐶}^2(ℝ)$ and ${𝑓}^{″}\ne 0$. Show that $𝑣={𝑓}^{\prime }(𝑢)$ is a solution to the Burgers equation

(ii) Now, suppose that $𝑢(𝑡,𝑥)$ is not a smooth solution to this conservation law but rather an entropy solution. Is then $𝑣={𝑓}^{\prime }(𝑢)$ an entropy solution to the Burgers equation?

We use the notation $[𝐴,𝐵]=𝐴𝐵-𝐵𝐴$ for the commutator of two operators $𝐴$ and $𝐵$.

(i) Suppose that an $𝑛\times 𝑛$ symmetric real-valued matrix $𝐿(𝑡)$ satisfies an ODE of the form

with some $𝑛\times 𝑛$ matrix $𝐴(𝑡)$. Show that the eigenvalues ${𝜆}_{𝑘}(𝑡)$ of the matrix $𝐿(𝑡)$ do not depend on $𝑡$. Is the same true for the eigenvectors of $𝐿(𝑡)$?

(ii) Consider a system of $𝑁$ particles located at positions ${𝑞}_{𝑘}(𝑡)$, $𝑘=1,\dots ,𝑁$, on the real line. Let us denote their momenta by ${𝑝}_{𝑘}(𝑡)$, $𝑘=1,\dots ,𝑁$. The Hamiltonian governing their motion is

and the equations of motion are

Let $𝐿(𝑡)$ be the symmetric tri-diagonal $𝑁\times 𝑁$ matrix with the diagonal entries

and the entries just above and just below the main diagonal equal to

All other entries of $𝐿(𝑡)$ are equal to zero. Find a matrix $𝑃(𝑡)$ such that the system can be written as

(iii) It follows from part (ii) of this problem that $Tr({𝐿}^{𝑘}(𝑡))$ does not depend on time for all $𝑘\ge 1$. Compute these conserved quantities explicitly in terms of ${𝑝}_{𝑗}$ and ${𝑞}_{𝑗}$ for $𝑘=1$, $𝑘=2$ and $𝑘=3$.

Consider the following dispersive regularization of the Burgers equation:

Find a third order in ${𝜕}_{𝑥}$ differential operator $𝐴(𝑡)$ of the form

and an operator

with the coefficients ${𝑎}_{𝑗}(𝑡,𝑥)$, $𝑗=0,1,2,3$ that depend on $𝑢(𝑡,𝑥)$, and fixed ${𝑚}_{1,2}\in ℝ$, so that this equation can be written as

Fix $0<𝑟<𝑅$ and let $𝜏(𝑥;𝜔)$ be the time it takes for the standard Brownian motion $𝐵(𝑡;𝑥)$ starting at the point $𝑥\in (𝑟,𝑅)$ to exit the interval $[𝑟,𝑅]$.

(i) Find $\mathbb{P}(𝐵(𝜏(𝑥;𝜔))=𝑟)$.

(ii) Find $𝑢(𝑡,𝑥)=\mathbb{P}(𝜏(𝑥;𝜔)>𝑡)$, for a given $𝑡>0$.

(iii) What is $𝔼(𝜏(𝑥;𝜔))$?

Let ${𝐵}_{𝑡}$ be the standard Brownian motion.

(i) Find all $𝑎\in ℝ$ and $𝑏\in ℝ$ such that

is a martingale.

(ii) For which $𝑚\in \mathbb{N}$ does there exist a deterministic function ${𝑔}_{𝑚}(𝑡)$ such that ${𝐿}_{𝑚}(𝑡)={𝐵}_{𝑡}^{𝑚}-{𝑔}_{𝑚}(𝑡)$ is a martingale?