By Walter Thirring

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M. for the investigation of nonstationary processes in the vicinity of the surface of transition through the speed of sound in Laval nozzles when the dimensions and form of the critical cross section change with time sufficiently rapidly,” [505, p. 939]. 10) and admits the following exact Ryzhov–Shefter solutions, 1959 (we keep the original notation from [505]): ϕ = λ(t)x + 1 2 A(t)[x − (t)]2 + h 1 (ϑ, t)[x − (t)]r 2 + h 2 (ϑ, t)r 4 . 11) 1 Linear Invariant Subspaces: Examples 5 The expansion coefficients solve the following PDE system: 1 λt + 2 Aλ = A t , 2 A + A2 = h 1ϑϑ + 4h 1 , t 2h 1t + h 1 A = h 2ϑϑ + 16h 2 .

1), in the sense that for any u ∈ W3 , F[u] ∈ W3 (or F[W3 ] ⊆ W3 ). This invariance of W3 under F is understood, as in standard linear algebra. 1) that is posed in IR 3 . 3) yields a system of linear elliptic PDEs for u 0 and u 1 that can be studied by standard techniques. 4) with a given nonlinear function f (u). The typical power nonlinearity is f (u) = ±u p , with the exponent p > 1, or f (u) = |u| p−1 u for solutions u of changing sign. In elliptic theory, two classes of problems were most popular: (i) the Dirichlet problem in a bounded domain ⊂ IR N , with u = 0 on the boundary ∂ , and (ii) the problem in the whole space IR N .

For n ≤ 5, the proof is straightforward by plugging the finite sum expansion u = C1 + C2 x + ... 56) and equating the coefficients of the expansion of F[u], corresponding to higher-degree terms x l with l ≥ n, to zero. , are suitable for this analysis. 8). A similar approach applies to other propositions presented below for various operators and subspaces. 57) + b6 u x x u x u + b5 u x x u 2 + b4 (u x )3 + b3 (u x )2 u + b2 u x u 2 + b1 u 3 only for the following three cases: (i) n = 2 with an 8D space spanned by F1 [u] = (u x x )3 , F2 [u] = u x (u x x )2 , F3 [u] = u(u x x )2 , F4 [u] = (u x )2 u x x , F5 [u] = uu x u x x , F6 [u] = u 2 u x x , F7 [u] = (u x )3 , F8 [u] = u(u x )2 ; (ii) n = 3 with a 6D space spanned by F1 [u] = (u x x )3 , F2 [u] = u x (u x x )2 , F3 [u] = u(u x x )2 , F4 [u] = (u x )2 u x x , F5 [u] = u x [2uu x x − (u x )2 ], F6 [u] = u[2uu x x − (u x )2 ]; (iii) n = 4 with a 2D space spanned by F1 [u] = (u x x )3 and F2 [u] = u x x uu x x − 23 (u x )2 .