Section A
Let us consider the following Cauchy problem associated to a first order PDE \[\begin{equation} \label{eq:q1} \left\{ \begin{array}{ll} x_2\partial_{x_1}u(x_1,x_2) = 1, & (x_1,x_2)\in\mathbb{R}^2,\\ u(0,x_2) = 0, & x_2\in\mathbb{R}. \end{array} \right. \end{equation}\]
Identify the leading vector field, the Cauchy data and the Cauchy curve.
Are the points on the Cauchy curve characteristic or non-characteristic? Justify your answer.
Using the method of characteristics, solve the problem in \(\eqref{eq:q1}\). Give the domain of definition of the solution.
Consider the Cauchy problem for Burgers’ equation \[\begin{equation} \label{eq:q2} \left\{ \begin{array}{ll} \partial_{t}u(x,t)+\frac12\partial_x(u^2(x,t)) = 0, & (x,t)\in\mathbb{R}\times (0,+\infty),\\ u(x,0) = u_0(x), & x\in\mathbb{R}, \end{array} \right. \end{equation}\] where \(u_0:\mathbb{R}\to\mathbb{R}\) is given.
Let \(u_0(x)=\frac17 x^{7}\). Show that \(\eqref{eq:q2}\) has a global in time classical solution.
Let \(u_0(x) = \sin(x)\). Write down the definition of the critical time \(t_c\) (until when we can guarantee the existence of a classical solution to \(\eqref{eq:q2}\)) associated to this initial datum. Show that \(t_c\le 1\).
In this problem we consider harmonic function on the unit ball in \(\mathbb{R}^3\), \(B_1(0)\).
Using the fact that the Laplacian in spherical coordinates is given by \[\Delta \psi = \frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2 \frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2\sin\theta}\left( \sin\theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^2\sin^2\theta}\frac{\partial^2 \psi}{\partial \phi^2}\] show that if \(u\in C^2\left( \overline{B_1\left( 0\right)}\right)\) is radial (i.e. only depends on \(r\) in spherical coordinates) and harmonic in \(B_1\left( 0\right)\) then it must be constant.
Show that there exists no radial solution in \(C^2\left( \overline{B_1(0)}\right)\) to the equation \[\nonumber \left\{ \begin{array}{ll} -\Delta u(\vec{x})=0, &\vec{x}\in B_1\left( 0\right),\\ u(\vec{x})=f(\vec{x}), & \vec{x}\in \partial B_1\left( 0\right), \end{array} \right.\] for \(f(\vec{x})=x_1^2\).
Consider the heat-like equation \[\begin{equation} \label{eq:heat_like} \left\{ \begin{array}{ll} u_t-u_{xx} +cu =0, & \left( x,t\right)\in\mathbb{R}\times\left( 0,+\infty\right),\\ u(x,0)=g(x), & x\in\mathbb{R}, \end{array} \right. \end{equation}\] where \(c\in\mathbb{R}\) is a fixed constant and \(g\in C_c\left( \mathbb{R}\right)\).
Define \(v(x,t)=e^{ct}u(x,t)\). Show that \(v(x,t)\) solves the heat equation \[\begin{equation} \nonumber %\label{eq:heat} \left\{ \begin{array}{ll} v_t-v_{xx} =0, & \left( x,t\right)\in\mathbb{R}\times\left( 0,+\infty\right),\\ v(x,0)=g(x), & x\in\mathbb{R}. \end{array} \right. \end{equation}\]
Show that there exists a solution to \(\eqref{eq:heat_like}\) that satisfies \[\sup_{x\in\mathbb{R}}\left\lvert u(x,t)\right\rvert \leq e^{-ct}\left\lVert g\right\rVert_{L^\infty\left( \mathbb{R}\right)}.\] You may use the following inequality without proof: For any \(f\in L^1\left( \mathbb{R}\right)\) and \(g\in L^\infty\left( \mathbb{R}\right)\) we have that \[\left\lvert\int_{\mathbb{R}}f\left( x-y\right)g(y)\right\rvert \leq \left\lVert f\right\rVert_{L^1\left( \mathbb{R}\right)}\left\lVert g\right\rVert_{L^\infty\left( \mathbb{R}\right)},\qquad \forall x\in\mathbb{R}.\]
Section B
We consider the following conservation law \[\begin{equation} \label{eq:q5} \left\{ \begin{array}{ll} \partial_{t}u(x,t)-u(x,t)\partial_x u(x,t) = 0, & (x,t)\in\mathbb{R}\times (0,+\infty),\\ u(x,0) = u_0(x), & x\in\mathbb{R}. \end{array} \right. \end{equation}\] [Notice that this is not Burgers’ equation.]
Suppose that \(u_0\) is bounded, differentiable with bounded derivative. Give a formula of the critical time \(t_c\), for which we know that \(\eqref{eq:q5}\) has a classical solution on \(\mathbb{R}\times(0,t_c)\).
Let \(u_0(x)=-\arctan(x)\). Show that in this case \(\eqref{eq:q5}\) has a global in time classical solution.
Let \(u_0\) now be given by \[u_0(x)=\left\{ \begin{array}{ll} 0, & x<0,\\ 1, & x\ge 0. \end{array} \right.\] By drawing the characteristics, show that there is instantaneous crossing of characteristics. Find a shock that satisfies the Rankine–Hugoniot condition. Give the expression of the weak solution in this case.
We aim to solve the following problem by the method of characteristics \[\begin{equation} \label{eq:q6} \left\{ \begin{array}{ll} \partial^2_{xx}u - 3\partial^2_{xy}u+2\partial^2_{yy}u=0, & (x,y)\in\mathbb{R}^2,\\ u(1,y) = g(y), & y\in\mathbb{R},\\ \partial_xu(1,y) = h(y), & y\in\mathbb{R}, \end{array} \right. \end{equation}\] where \(g,h:\mathbb{R}\to\mathbb{R}\) are given smooth functions.
Identify the Cauchy data and the Cauchy curve in the above problem.
Rewrite the PDE in \(\eqref{eq:q6}\) as a system of two linear first order PDEs. [Hint: think about the algebraic relation \((a-b)(a-2b)=a^2-3ab+2b^2,\ (a,b\in\mathbb{R})\).]
By solving the two first order PDEs arising from [qAin6.2] using the method of characteristics, find the solution to \(\eqref{eq:q6}\).
Let \(\Omega\) be an open bounded set with smooth boundary in \(\mathbb{R}^n\) and let \(u_1\) and \(u_2\) be \(C^2\left( \Omega\right)\cap C\left( \overline{\Omega}\right)\) solutions to Poisson equation \[\nonumber \left\{ \begin{array}{ll} -\Delta u_i(\vec{x})=f\left( \vec{x}\right), & \vec{x}\in \Omega,\\ u_i(\vec{x})=g_i(\vec{x}), & \vec{x}\in \partial \Omega, \end{array} \right.\] \(i=1,2\), where \(f\in C^1\left( \overline{\Omega}\right)\) and \(g_1,g_2\in C\left( \partial \Omega\right)\).
Show that for any \(\vec{x}\in\overline{\Omega}\) \[u_2(\vec{x})-u_1(\vec{x}) \leq \max_{\vec{x}\in \partial \Omega}\left( g_2(\vec{x})-g_1(\vec{x})\right).\]
Show that \[\max_{\vec{x}\in\overline{\Omega}}\left\lvert u_2(\vec{x})-u_1(\vec{x})\right\rvert \leq \max_{\vec{x}\in \partial \Omega}\left\lvert g_2(\vec{x})-g_1(\vec{x})\right\rvert.\]
For \(n\in\mathbb{N}\) let \(u_n\in C^2\left( \Omega\right)\cap C\left(
\overline{\Omega}\right)\) solve the system \[\nonumber
\left\{
\begin{array}{ll}
-\Delta u_n(\vec{x})=f\left( \vec{x}\right), &
\vec{x}\in \Omega,\\
u_n(\vec{x})=g_n(\vec{x}), & \vec{x}\in \partial
\Omega,
\end{array}
\right.\] and let \(u\in
C^2\left( \Omega\right)\cap C\left( \overline{\Omega}\right)\)
solve the system \[\nonumber
\left\{
\begin{array}{ll}
-\Delta u(\vec{x})=f\left( \vec{x}\right), & \vec{x}\in
\Omega,\\
u(\vec{x})=g(\vec{x}), & \vec{x}\in \partial \Omega.
\end{array}
\right.\] Show that if \(\left\lbrace
g_n\right\rbrace_{n\in\mathbb{N}}\) converges uniformly to \(g\) on \(\partial
\Omega\) then \(\left\lbrace
u_n\right\rbrace_{n\in\mathbb{N}}\) converges uniformly to \(u\) on \(\overline{\Omega}\).
Recall that we say that a sequence of functions \(\left\lbrace
f_n\right\rbrace_{n\in\mathbb{N}}\) in \(C\left( K\right)\) converges uniformly to
\(f\in C\left( K\right)\) if \[\sup_{x\in K}\left\lvert
f_n(x)-f(x)\right\rvert\underset{n\to\infty}{\longrightarrow}0.\]
Let \(u\in L^1\left( \mathbb{R}\right)\cap L^2\left( \mathbb{R}\right)\) be a classical solution to the equation \[\nonumber \left\{ \begin{array}{ll} u_t + ku_{xxxx}=0,& \left( x,t\right)\in \mathbb{R}\times\left( 0,+\infty\right),\\ u(x,0)=f(x),& x\in\mathbb{R}, \end{array} \right.\] where \(k>0\) is a fixed constant and \(f\) is a smooth function on \(\mathbb{R}\) that belongs to \(L^1\left( \mathbb{R}\right)\cap L^2\left( \mathbb{R}\right)\).
Show that \(\widehat{u}\), the Fourier transform of \(u\) in the \(x-\)variable, satisfies \[\widehat{u}(\xi,t)=\widehat{f}(\xi)e^{-k\xi^4 t}.\]
Using the fact that the Fourier transform preserves the \(L^2\) norm (Plancherel’s identity) as well as the fact that it is in \(L^\infty\) to show that \[\left\lVert u(\cdot,t)\right\rVert^2_{L^2\left( \mathbb{R}\right)} \leq \frac{\left( \int_{\mathbb{R}}e^{-x^4}dx\right)^{\frac{1}{2}}}{\displaystyle\sqrt[8]{4kt}}\left\lVert f\right\rVert_{L^1\left( \mathbb{R}\right)}\left\lVert f\right\rVert_{L^2\left( \mathbb{R}\right)}.\]
For \(n\in\mathbb{N}\) we define \(f_n:\mathbb{R}\to\mathbb{R}\) as \(f_n(x)=\sqrt{x^2+1/n}\).
Show that as \(n\to+\infty\), the sequence \((f_n)_{n\in\mathbb{N}}\) converges uniformly to \(f:\mathbb{R}\to\mathbb{R}\) given by \(f(x)=|x|\).
Compute the pointwise limit of the sequence \((f'_n)_{n\in\mathbb{N}}\) as \(n\to+\infty\). Can this limit be the uniform limit of the sequence? Justify your answer.
Show that as \(n\to+\infty\), the sequence \((f''_n)_{n\in\mathbb{N}}\) converges in the sense of distributions to \(2\delta_0\), where \(\delta_0\) is the Dirac mass concentrated at \(0\).