Suppose that \(u\) is smooth and solves the heat equation : \[\partial_t u - \Delta u = 0 \quad \text{in} \quad \mathbb{R}^n\times (0,+\infty).\] Show that \(v(x,t) = x\cdot\nabla u(x,t) + 2 t \partial_t u(x,t)\) solves also the heat equation.
Hint: note that \(u(\lambda x, \lambda^2 t)\) solves the heat equation for all \(\lambda >0.\)
Exercise 2
We suppose \(n=1\) and set \(u(x,t) = v\left(\frac{x^2}{t} \right).\)
Show that \(\displaystyle\partial_t u - \partial_x^2 u = 0\) if and only if \[\label{eq:heat}
4zv''(z) + (2+z) v'(z) =0 \qquad (z>0).\]
Show that the general solution to [eq:heat] is \[v(z) = a \int_0^z e^{-s/4} s^{-1/2} ds + b.\]
Differentiate \(\displaystyle v\left(\frac{x^2}{t} \right)\) with respect to \(x\) and select the parameter \(a\) properly to obtain \(\Phi\) the fundamental solution for the heat equation in the case \(n=1.\)
Hint: The fundamental solution for the heat equation is given in (4) in the manuscript:
\[\Phi(x,t) = \begin{cases}
\frac{1}{(4\pi t)^{n/2}} e^{-\frac{\vert x \vert^2}{4t}} & x \in \mathbb{R}^n, t>0, \\
0 & x\in \mathbb{R}^n, t<0.
\end{cases}\]
Exercise 3
We say that \(u\) is a subsolution of the heat equation in \(\Omega \times (0, T)\) if \[\partial_t u - \Delta u \leq 0 \quad \text{in} \; \Omega\times (0,T).\]
Suppose that \(u\) is solution of the heat equation in \(\Omega \times (O,T)\) and \(\phi\) a smooth convex function. Show that \(\phi(u)\) is a subsolution of the heat equation.
Prove that \(\displaystyle\vert \nabla u\vert^2 + u_t^2\) is also a subsolution when \(u\) is solution of the heat equation.
Exercise 4
Let \(b \in \mathbb{R}^n\) and \(c \in \mathbb{R}\) are constants. Write down an explicit formula for a function \(u\) solving the initial value problem \[\begin{cases}
\partial_t u + b\cdot \nabla u + c u =0 & \text{in} \; \mathbb{R}^n\times(0,+\infty), \\
u = g & \text{on} \; \mathbb{R}^n \times \{0\}
\end{cases}\]
Exercise 5
Let \(u\) be a smooth function and a triangular region defined in Figure 5.
Definition of the triangular region \(T.\)
Using the divergence theorem in the triangle T show that \[u(C) = \frac{1}{2} (u(A) + u(B))+ \frac{1}{2}\int_A^B u_t dx 1 +\frac{1}{2} \int \int_T (\partial_t^2 u -\partial_x^2 u) dx dt.\]
Hint: if \(\mathrm{div}w = \partial_x w_1 + \partial_t w_2\) note that \(\partial_t^2 u - \partial_x^2 u = \mathrm{div}(-partial_x u,\partial_t u).\)
Exercise 6
Show that the general solution of the partial differential equation \(\partial_{xy}^2 u =0\) is \[u(x,y) =F(x) + G(y)\] where \(F, G\) are arbitrary functions.
Setting \(\xi = x+t\) and \(y = x-t,\) show that \[\partial_t^2 u - \partial_x^2 u =0 \iff \partial_{\xi y}^2 u =0.\]
Recover the d’Alembert formula.
Exercise 7
Let \(u \in C^2(\mathbb{R}\times[0,+\infty))\) the solution to \[\begin{cases}
\partial_t^2 u - \partial_x^2 u = 0 & \text{in} \; \mathbb{R}\times (0,+\infty), \\
u = g, u_t = h & \text{in} \; \mathbb{R}\times \{0\}.
\end{cases}\] Suppose that the functions \(g\) and \(h\) have compact supports. The kinetic energy of \(u\) is given by \({\displaystyle k(t) = \frac{1}{2} \int_{\mathbb{R}}u_t^2(x,t) dx},\) its potential energy by \({\displaystyle p(t) = \frac{1}{2} \int_{\mathbb{R}} u_x^2(x,t) dx}.\)
Show that \(k(t) + p(t)\) is constant in \(t\) and \(k(t) = p(t)\) for large time \(t.\)
Hint: use the energy method and the d’Alembert formula.
Exercise 8 • Exercise 10 of the manuscript
Find divergence free functions in \(\mathbb{R}^n,\) i.e. some vector fields \(u\) such that \(\mathrm{div} u =0.\)
Show that they built an infinite dimensional space.