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**Solution of continuous nonlinear PDEs through order completion.**
*(English)*
Zbl 0821.35001

North-Holland Mathematics Studies. 181. Amsterdam: Elsevier Science B. V.. xvi, 432 p. (1994).

“Solutions of continuous nonlinear PDEs through order completion” is a book which claims to produce global and unique solutions to every \(m\)-th order nonlinear PDE \(T(x,D) U(x)= f(x)\), \(x\in \Omega\subset \mathbb{R}^ n\), whenever \(f\in C^ 0 (\Omega)\). Of course, we all learn early in our training that uniqueness is not to be expected unless one imposes additional initial or boundary conditions, but even then, global existence is usually elusive, as seen from the classical (ODE) example \({d\over dx} u(x)- u^ 2 (x)= 0\), \(u(0)=1\). This example is actually quoted in the book. The authors arrive at their “much stronger” existence and uniqueness results by relaxing the concept of solution to the utmost and by enforcing uniqueness by calling different solutions “equivalent” and lumping them together as one object (an equivalence class).

The main step is taken early in the book in Proposition 2.1, which says that \(\forall f\in C^ 0( \overline {\Omega})\) and \(\forall \varepsilon>0\) there is a nowhere dense set \(\Gamma_ \varepsilon \subset \Omega\) and a function \(U_ \varepsilon \in C^ m (\Omega \smallsetminus \Gamma_ \varepsilon)\) such that on \(\Omega \smallsetminus \Gamma_ \varepsilon\) holds: \(f- \varepsilon\leq T(x, D) U_ \varepsilon\leq f\). The proof of this result is by polynomial approximation, and the trouble is on the exception set \(\Gamma_ \varepsilon\), where the approximation is simply not required. “Solutions” of the general PDE are then constructed via “Dedekind order completion”, an abstract tool generalizing a classical way of obtaining the reals from the rationals. The drawback is that the objects called solutions do not solve everywhere; moreover, on the exceptional set they are not even required to be continuous! For the standard example given earlier, this means that “global existence” is obtained but forcing jumps into the solution whenever there is a danger of blowup. This seems to be the basic idea behind the method, but it is presented inside a general abstract setting.

Does it make sense from a physical point of view? It probably depends on the particular problem under consideration. For many equations, the failure to have a global (or unique) solution carries a lot of physical sense (blowup; bifurcations; shock wave formation), but for other problems (e.g., in control theory), jumps may be acceptable. This reviewer wishes that the authors had attempted a discussion of the physical meaning of their solution concept. For example, in shock wave formation and propagation for hyperbolic conservation laws (the Burgers’ equation is discussed as an example in this book in Chapter 16), discontinuities in the solutions arise quite naturally. The differential formulation at such discontinuities then causes difficulties, and the researcher must go to a more fundamental form of the equation, e.g., the integral form of a conservation law (which is equivalent to a weak formulation of the differential equation). The formation and behavior of a discontinuity then becomes a natural part of the solution, whereas in the general theory presented here, the discontinuities arise as an artifact of the approximation procedure.

The first 5 chapters of the book contain the basic theory, from the approximation result quoted above, given in Chapter 2, through the idea of Dedekind order completion (Chapter 3) to the concept of generalized solutions in the Dedekind order completion (Chapter 4) and the general global existence and uniqueness result in the sense described earlier (Chapter 5). Examples are given in Chapters 6, 8, 10, 11 and 16.

In Chapter 6, the reader gets a first idea what can be done with this theory: The authors use their results to obtain a certain nonexistence result (Prop. 6.1) if the r.h.s. of \(D_ t U(t, y)= f(y)\) is nondifferentiable on a dense subset \(S\subset \mathbb{R}\). Other, classical examples of nonexistence results are mentioned in this section. Chapter 7 is concerned with the interpretation and structural properties of Dedekind order solutions. Chapter 8, the beginning of part II of the book, presents the solution of Cauchy problems (in a straightforward way, by restricting the equivalence class of solutions to such functions which assume the right initial values). In Chapter 9, more abstract equations are considered, and Chapter 10 discusses what happens when the equations do have classical solutions. The relationship of the new approach to classical functional analytic methods is the topic of Chapter 12. Extensions of the theorem are discussed in Chapters 13 and 14. Part III of the book (Chapters 15 through 18) talks about group invariance properties of the new solution concept. It is in this part that semilinear hyperbolic systems and scalar conservation laws are discussed as examples.

This reviewer remains unconvinced that the new approach has really solved any of the hard classical problems in the theory of PDEs (many self- praising assertions of the authors nonwithstanding). Yet there is a lot of new mathematics here, for those who like abstraction. The future will tell whether this approach to PDEs is acceptable to physicists, engineers, biologists etc. As mentioned at the beginning of this review, nonexistence of a solution sometimes tells more than existence.

The main step is taken early in the book in Proposition 2.1, which says that \(\forall f\in C^ 0( \overline {\Omega})\) and \(\forall \varepsilon>0\) there is a nowhere dense set \(\Gamma_ \varepsilon \subset \Omega\) and a function \(U_ \varepsilon \in C^ m (\Omega \smallsetminus \Gamma_ \varepsilon)\) such that on \(\Omega \smallsetminus \Gamma_ \varepsilon\) holds: \(f- \varepsilon\leq T(x, D) U_ \varepsilon\leq f\). The proof of this result is by polynomial approximation, and the trouble is on the exception set \(\Gamma_ \varepsilon\), where the approximation is simply not required. “Solutions” of the general PDE are then constructed via “Dedekind order completion”, an abstract tool generalizing a classical way of obtaining the reals from the rationals. The drawback is that the objects called solutions do not solve everywhere; moreover, on the exceptional set they are not even required to be continuous! For the standard example given earlier, this means that “global existence” is obtained but forcing jumps into the solution whenever there is a danger of blowup. This seems to be the basic idea behind the method, but it is presented inside a general abstract setting.

Does it make sense from a physical point of view? It probably depends on the particular problem under consideration. For many equations, the failure to have a global (or unique) solution carries a lot of physical sense (blowup; bifurcations; shock wave formation), but for other problems (e.g., in control theory), jumps may be acceptable. This reviewer wishes that the authors had attempted a discussion of the physical meaning of their solution concept. For example, in shock wave formation and propagation for hyperbolic conservation laws (the Burgers’ equation is discussed as an example in this book in Chapter 16), discontinuities in the solutions arise quite naturally. The differential formulation at such discontinuities then causes difficulties, and the researcher must go to a more fundamental form of the equation, e.g., the integral form of a conservation law (which is equivalent to a weak formulation of the differential equation). The formation and behavior of a discontinuity then becomes a natural part of the solution, whereas in the general theory presented here, the discontinuities arise as an artifact of the approximation procedure.

The first 5 chapters of the book contain the basic theory, from the approximation result quoted above, given in Chapter 2, through the idea of Dedekind order completion (Chapter 3) to the concept of generalized solutions in the Dedekind order completion (Chapter 4) and the general global existence and uniqueness result in the sense described earlier (Chapter 5). Examples are given in Chapters 6, 8, 10, 11 and 16.

In Chapter 6, the reader gets a first idea what can be done with this theory: The authors use their results to obtain a certain nonexistence result (Prop. 6.1) if the r.h.s. of \(D_ t U(t, y)= f(y)\) is nondifferentiable on a dense subset \(S\subset \mathbb{R}\). Other, classical examples of nonexistence results are mentioned in this section. Chapter 7 is concerned with the interpretation and structural properties of Dedekind order solutions. Chapter 8, the beginning of part II of the book, presents the solution of Cauchy problems (in a straightforward way, by restricting the equivalence class of solutions to such functions which assume the right initial values). In Chapter 9, more abstract equations are considered, and Chapter 10 discusses what happens when the equations do have classical solutions. The relationship of the new approach to classical functional analytic methods is the topic of Chapter 12. Extensions of the theorem are discussed in Chapters 13 and 14. Part III of the book (Chapters 15 through 18) talks about group invariance properties of the new solution concept. It is in this part that semilinear hyperbolic systems and scalar conservation laws are discussed as examples.

This reviewer remains unconvinced that the new approach has really solved any of the hard classical problems in the theory of PDEs (many self- praising assertions of the authors nonwithstanding). Yet there is a lot of new mathematics here, for those who like abstraction. The future will tell whether this approach to PDEs is acceptable to physicists, engineers, biologists etc. As mentioned at the beginning of this review, nonexistence of a solution sometimes tells more than existence.

Reviewer: R.Illner (Victoria)