a hyperbolic initial value problem
... with a radiation operator definition at t=0
... and the Ritz-Galerkin method for the wave equation
There is a conceptual difference between elliptic and parabolic problem versus hyperbolic problems.
The Maxwell equations build a symmetric hyperbolic system. The decoupled Maxwell equations consist of two linear wave equations. The telegraph equation is a generalized wave equation, where different choises of the underlying "main parameter a) leads to hyperbolic, elliptic and parabolic differential equations.
The Yang-Mills equations are a nonlinear generalization of the Maxwell equations which are semi-linear.
The Euler and Einstein equations are quasi-linear, but not semi-linear hyperbolic equations. Even the simplest linear hyperbolic equation (the wave equation) shows different ('not optimal') properties of shift theorems in standard Sobolev spaces.
At the same time, there is the Courant conjecture, which makes the 2D and 4D space-time worlds unique compared to all other dimensions n (Courant-Hilbert, Methods of Mathematical Physics) p. 763:
"Families of spherical waves for arbitrary time-like lines exist only in the case of two or four variables, and then only if the differential equation is equivalent to the wave equation".
In this context we also note that in current "quantum radiation theories" the Huygens principles is neglected.
With respect to hyperbolic problems the adequate energy Hilbert space cannot be more regular than the proposed H(1/2) (energy) Hilbert space for the NSE solution. We provide a counter example that it is most likely that any H(a) for any a>0 is not sufficient. An alternative (“exponential decay”) inner product is proposed with ‘optimal’ shift theorem behavior of the d’Alembert operator. The prize to be paid is an extension of current Hilbert scale inner products with polynomial decay coefficients to an inner product with exponential decay.
We present an 'optimal order' shift theorem for the wave equation with respect to this alternative ("hyperbolic") energy norm. At the same time we emphasis that there is only one quantum potential energy “measure” (as proposed in http://www.fuchs-braun.com/ ), which is the quantum potential energy norm.
We note the elegant role of the proposed NSE-H(1/2) energy norm in universal Teichmüller theory.
We provide a characterization of the 4-dimensional Minkowski space which is about the fact that the differential of the Fourier transform of the uniform distribution of unit mass over the unit sphere at the origin vanishes.
Following the same concept as for the parabolic (heat equation) case (Nitsche/Wheeler) we sketch a proof of the L(infinite) boundedness of the Ritz operator in case of finite element approximation spaces for the solution of the wave equation. The analysis restricts to the space dimension m=1 and m=3. The required assumptions to the finite element approximation spaces are linear resp. quadratic finite elements in order to enable the Nitsche duality technique.
"the objective world has only been constructed at the price of taking the self, that is, mind, out of it, remaking it; obviously, therefore, it can neither act on it nor be acted on by any of ist parts. ... If this problem of the action of mind on matter cannot be solved within the framework of our scientific representatiion of the objective world, where and how can it be solved?" - E. Schrödinger, Mind and Matter