Problem set 1: Well-posedness of IBVP#

Topics covered: Various PDEs and boundary conditions.

Note: If you miss the problem solving class, you have to submit solutions to the following exercises in Studium before the deadline: 1 a)-d) and 2 a)-c)

Exercise 1#

Consider the scalar initial-boundary value problem (IBVP)

\[\begin{split} \begin{array}{lll} c\,u_t=a\,u_{x} +(b\,u_{x})_x + d\,u_{xxx} , & 0\le x \le L, & t\ge 0,\\ \mathcal{L}_lu=g_l , &x=0,& t\ge 0,\\ \mathcal{L}_ru=g_r, &x=L,& t\ge 0,\\ u=f,& 0\le x \le L, & t= 0,\\ \end{array} \end{split}\]

where \(c=c(x)>0\) is a real-valued function; \(b=b(x)\) is possibly complex-valued; and \(a\) and \(d\) are (possibly complex-valued) constants.

a) Consider the case \(d=0\). What are the requirements on \(a\), \(b\), and \(c\) for the PDE to be well-posed, disregarding the boundary conditions? That is, you may assume periodic boundary conditions. Hint: Use the energy method.

b) For \(d=0\), what are the requirements for the PDE to conserve some energy? Again consider periodic boundary conditions.

c) For \(d=0\), derive at least two sets of well-posed boundary conditions. That is, find two different operators \(\mathcal{L}_l\) (and \(\mathcal{L}_r\)) that yield a well-posed IBVP.

d) Consider the case \(a=c=1\), \(d=0\), \(b=10\). Describe the expected behaviour of the solution.

e) Now consider \(d\neq0\). What are the requirements for the PDE to be well-posed with periodic boundary conditions? Hint: the term \(d\,u_{xxx}\) requires integrating by parts twice.

f) For \(d\neq0\), derive one set of well-posed boundary conditions. Hint: You will need 3 conditions in total due to the term \(d\,u_{xxx}\).

Exercise 2#

Consider the IBVP

\[\begin{split} \begin{array}{lll} {\bf C}{\bf u}_{t}={\bf A}{\bf u}_x +{\bf B}{\bf u}+{\bf F}, & 0\le x \le L, & t\ge 0,\\ \mathcal{L}_l{\bf u}=g_l , &x=0,& t\ge 0,\\ \mathcal{L}_r{\bf u}=g_r, &x=L,& t\ge 0,\\ {\bf u}={\bf f},& 0\le x \le L, & t= 0,\\ \end{array} \end{split}\]

where \({\bf F}={\bf F}(x,t)\) is the forcing function, \({\bf f}\) is the initial data, \({\bf A}={\bf A}^*\) is a constant matrix, \({\bf B}\) and \({\bf C}\) are variable-coefficient matrices, and \(\mathbf{C} = \mathbf{C}^* > 0\). \({\bf A}\) has the structure

\[\begin{split} {\bf C}=\begin{bmatrix} c_1(x)&0 & 0\\ 0&c_2(x)&0\\ 0&0&c_3(x)\\ \end{bmatrix}\;, \quad {\bf A}=\begin{bmatrix} 2&1 & 0\\ 1&0&2\\ 0&2&\alpha\\ \end{bmatrix}\;, \end{split}\]

where \(c_i \in \mathbb{R}\) and \(\alpha\) is a real constant.

a) Use the energy method to derive an energy rate for the IBVP with \(\mathbf{F}=0\). Under which conditions can you show that the PDE with periodic boundary conditions is well-posed?

b) How many boundary conditions should be prescribed at each boundary in the cases \(\alpha=0\), \(\alpha=-8\) and \(\alpha = -10\)?

c) Consider the case \(\alpha=0\). Derive at least one set of well-posed boundary conditions. You may assume that the solution is real-valued.