Comparing solutions of PDE problem with different initial conditions

Question

My question(s) is about what happens with the solution of the problem if we change initial conditions.

Let's say we have a PDE problem:

$$ (1) \hspace{0.5cm} u_t+f(u)_x=0 $$ $$ (2) \hspace{0.5cm} u(x,0)=u_0 (x) $$

where $x \in A \subseteq \mathbb{R}$, $t \in [0,T]$ and $u \in \mathbb{R}^n$.

Here we have equation or a system of equations written in the conservation form. (1),(2) is usually called the Cauchy problem. If we change initial condition (2) into:

$$(3) \hspace{0.5cm} u(x,0)= \begin{cases} u_l, x<0 \\[2ex] u_r, x>0 \end{cases} $$

where $u_l$ and $u_r$ are constants, (1),(3) is usually called the Riemann problem.

I have three questions.

1. What could we say or how we could compare solutions of (1), (2) with the solution (1), (3)? So here we compare the Cauchy problem and the Riemann problem.

2. What if we compare two problems of the same type i.e. we have $u_{01}$ and $u_{02}$ (both type (2) or both type (3))? So here we compare two Cauchy problems or two Riemann problems.

3. What if we have initial conditions (3) and initial conditions in (2) which originated from (3). For example, just mollify (3) with a Friedrich mollifier and we get the form (2) then)? So here we have the Cauchy problem and the Riemann problem but they have some connection.

Solutions could be strong or weak in the PDE sense.

I think that at least in some of these three cases some kind of energy estimate could be useful.

I am also interested in the cases where the system is not in the conservation form but that is another topic.

Please share your thoughts. And also if anyone knows some papers/books that deal with this kind of problems write it down.

EDIT: I'll give a one example for the question number one. Let's say that $(1)$ is the inviscid Burgers equation ($u_t + u u_x =0$) and that we have initial conditions given with $(3)$. One type of weak solution for the Riemann problem of the inviscid Burgers euqation is shock wave (when $u_l > u_r$):

$$(4) \hspace{0.5cm} u(x,t)= \begin{cases} u_l, x<s \cdot t \\[2ex] u_r, x>s \cdot t \end{cases}$$

Here $s$ is the speed of the discontinuity. If we apply to $(3)$ convolution with a mollifier $\phi_{\sigma}$, we get initial condition of type $(2)$. Now we have the Cauchy problem for the inviscid Burgers equation. Could we "imitate" the Riemann problem and say that the Cauchy problem has a solution that is given with:

$$(5) \hspace{0.5cm} u(x,t)= \begin{cases} u_l, x<s \cdot t-\sigma \\[2ex] h, s \cdot t-\sigma <x< s \cdot t + \sigma \\[2ex] u_r, x>s \cdot t + \sigma \end{cases}$$

where $h$ is some function that represent the mollified part. This would be the "mollified initial data that move in time".

Answer 1

I'm not sure this really qualifies as an answer (there's not much here beyond notation really), but one general way of thinking about this sort of problem is as follows. Sorry if any of this seems a bit basic; I am in no way an expert in this area. All I'm going for here is a fleshing out of the general shape of how I prefer to regard this sort of problem.

(1) Decide what sorts of functions/distributions you're willing to accept as solutions of your problem; you'll generally want this to an open subset $U$ of an appropriate topological vector space.

(2) Think of your problem as defining a (possibly nonlinear) operator $P:U\longrightarrow V$ into some open subset $V$ of another TVS; in this case, the $P$ you'd want might be something like:

$$ P(u) = \left( \begin{array}{c} u_t + f(u)_x \\[.5pc] Bu\end{array} \right) $$

with $(Bu)(x,t)=u(x,0)$ and $V$ an open subset of suitable product space. Solutions to the evolution equation (plus initial condition) are now just points $u\in U$ for which $P(u) = v$ with $v = (0,u_0)$ (sorry for the mix of row/column vectors).

(3) Assumning $U$ and $V$ are small enough for your problem to be sufficiently well posed as to admit a local inverse $Q:V'\longrightarrow U'$ (so that $QP$ and $PQ$ are the identities on $U'$ and $V'$ respectively), the $u$ corresponding to a given $u_0$ is

$$ Q\left( \begin{array}{c} 0 \\u_0\end{array} \right) $$

and this remains the case provided we vary $u_0$ in such a way that $(0,u_0)$ stays in $V'$ (the domain of definition of the local inverse). If the operator $P$ is suitably smooth, versions of the inverse function theorem may be available, and may even give you smooth local inverses (in which case your $u$ will depend `smoothly' on $u_0$).

If you're looking for references in this direction, maybe concentrate your search on books/notes which mention well posedness, especially as applied to nonlinear operators. Books with nonlinear analysis in the title may also have more to say on this side of things than standard PDE texts.