Existence and uniqueness of smooth solutions

2.1 Existence and uniqueness of smooth solutions

Let $M 0$ be the space of functions of the form:

1 ∫ ∞ iω⋅x ∞ n f(x) := (2π)n∕2 e ϕ(ω )dω, ϕ(ω) ∈ C0 (ℜ ). −∞

Since the integral exists at each point, it can be differentiated inside the integral sign where it gives another compactly supported integrand, thus these functions are smooth ( $C ∞(ℜn )$ ). What’s more, since the Paley–Wiener theorem holds, they are analytic.

To this space also belongs the function,

1 ∫ ∞ u(x,t) := ----n∕2 eiω⋅xeP(iω)tϕ(ω)dω, ϕ(ω ) (2π) − ∞

This function is smooth not only along space directions, but also along time directions. In fact, it is straightforward to check that the function satisfies the evolution equation above for the initial condition $f (x )$ . Thus we see that initial data in $M0$ always produces solutions which at each constant time slice are in $M0$ . In fact, defining a $M0$ solutions as:

Definition 2 A function $u (x,t)$ is a $M0$ solution if:

i) $u(⋅,t) ∈ M0$ for all $t ≥ 0$ ;

ii) its Fourier transform, $ˆu(ω,t)$ , is continuous, and vanishes for $|ω | > k$ where $k$ is some constant independent of $t$ ;

iii) $u$ is a classical solution; that is $u t$ exists and $u$ satisfies the equation at each point $(x,t)$ .

a direct application of the uniqueness of the Fourier representation for smooth functions shows:

Lemma 1 Given a constant coefficients linear evolution equation, for each initial data in $M0$ there exists a unique $M0$ solution and it is given by the above formula.

Thus we see that there are plenty of smooth solutions, whatever the system is. But it was realized by Hadamard [37 ] that there were not enough solutions, since the space $M0$ is not closed. Furthermore, in general there are no topologies on the space of initial data, and of solutions for which solutions depend continuously on initial data. Lack of continuity of solutions with respect to their initial data would not only imply lack of predictability from the physical standpoint, for all data are subject to measurement errors, but also lack of realistic possibilities of numerically computing solutions, due to truncation errors. Thus it is important to characterize the set of equations for which continuity holds. There are several possibilities for the choice of the topologies for the spaces of initial data and of solutions. Here we restrict consideration to those which have been more prolific with respect to results and generalizations to non-linear, non-constant coefficient equations systems.

Definition 3 A system of partial differential equations is called well posed if there exists a norm (usually a Sobolev norm) and two constants, $k$ , $α$ , such that for all initial data in $M0$ and all positive times,

αt ||u(t,⋅)|| ≤ ke ||f(⋅)||.

Remarks:

It is possible to define weaker conditions for well posedness in which the norm for the initial data is different (weaker) than the norm for the solution. This is unsatisfactory for equations in which there is no preferred time direction. Besides, in general it produces results which are not robust under lower order term (in differentiation) perturbations of the equations, and so do not generalize to variable coefficients or non-linear equations.
Defining continuity through norms is a limitation and rules out certain more general types of hyperbolicity conditions, in particular those due to Leray and Ohya [50 , 51 ], see more below.
Well posedness and linearity imply that we can extend by continuity the set of solutions to all those which are generated by initial data on the completion generated by $M0$ on the given norm.

Theorem 1 A system is $L2$ well posed if and only if there exists constants $k$ and $α$ such that for all positive times,

P(iω)t αt n |e | ≤ ke ∀ ω ∈ ℜ ,

where the above norm is the usual operator norm on matrices.

If a system is well posed for the $2 L$ norm, [recall that the $2 L$ norm of a function is the square root of the integral of its square], then it is well posed for any other Sobolev norm, (as follows from the above theorem), since the constants are independent of $ω$ . The above theorem reduces the problem of well posedness to an algebraic one which we further refine in the following theorem:

Theorem 2 [Kreiss [47 ] The following conditions are equivalent:

i) The system is $L2$ well posed.

ii) There exist constants $k$ , and $α$ , and a positive definite Hermitian form $H (ω)$ such that:

−1 ⋆ n k I ≤ H (ω ) ≤ kI and H (ω)P (iω) + P (iω)H (ω ) ≤ 2αH (ω) ∀ω ∈ ℜ .

This result is central to the theory. The proof that ii) implies i) is simple and follows directly from the inequality:

d-(ˆu,H (ω )ˆu) = (ˆu,H (ω)P (iω) + P⋆(iω)H (ω )ˆu) ≤ 2α(ˆu,H (ω )ˆu), dt

that is, from the construction of an energy norm. We see that for any well posed problems this special energy norm can be constructed, so one can always attempt to approach the problem by trying to find, usually with the help of the physics behind the problem, the correct energy norm. Condition ii) is usually referred to as the semiboundedness of the operator $P(D )$ with respect to the norm $H$ (induced on functions in $ℜn$ by Fourier Transform).