ABSTRACT
Here we demonstrate an application of another classic approach to nonlinear problems, which, in the present case of an unclear entropy nature of solutions of NDEs and the open uniqueness problem, gives a certain partial insight into both. We mean the Bubnov-Galerkin method, which was the most widely used approach for constructing weak solutions via finite-dimensional approximations and passing to the limit; see Lions [276] with many applications therein. Thus, by this classic theory of nonlinear problems, under the assumption
(161) and others, if necessary, let us perform a standard construction of a compactly supported (for simplicity) solution, using a basis {ψk} of eigenfunctions of the regular linear operator P2 = D
2 x < 0 with the Dirichlet
boundary conditions,
ψ′′ = λkψ, ψ(±L) = 0 =⇒ λk ∼ −k2, and u = 0 at x = L. (178) As an alternative, it is curious that, for our purposes, it is possible (and more convenient for some reasons) to use the eigenfunction set of the operator P4 = −D4x < 0 again with the Dirichlet conditions
ψ = ψx = 0 at x = ±L. Special Bubnov-Galerkin bases associated with higher-order operators P6 = D6x < 0 also may be convenient; see applications to third-order linear dispersion equations in [267]. In all these self-adjoint cases, the eigenfunctions form a complete and closed set in L2; see classic theory of ordinary differential operators in Naimark [309, p. 89]. On the other hand, looking for a more natural choice of the third-order
operator P3 = D 3 x for the Bubnov-Galerkin approximation of (1) will cause a
difficult problem, since, for the third-order PDE with the principal operator as in (1),
ut = a(x, t)uxxx + ... (a → u) (179) with a > 0, a proper setting for the IBV problem includes the Dirichlet conditions (see Faminskii [120] for details and a survey)
u = ux = 0 at x = −L and u = 0 at x = L. (180) For a < 0, the boundary conditions must be swapped, so that the proper setting of the problem depends on the unknown sign of solutions. Here, the fact that P3 = D
3 x is not self-adjoint is not essential, since, relative to the adjoint
basis {ψ∗k}, the closure and completeness of the bi-orthonormal generalized eigenfunction sets remain valid. Actually, the choice of linear operators P2 = D
2 x, P4 = −D4x, or others, is
not of principal importance, when looking for compactly supported solutions
u(x, t) ∈ C∞0 ((−L,L)) for all t ∈ [0, 1]. (181)
and
It should be noted that control of the finite propagation property in (1) is difficult and is an essential part of our further analysis. For instance, we also can fix periodic boundary conditions that are always regular [309, Ch. 2] (it is curious that (180) are not). Thus, we construct a sequence {um} of approximating Bubnov-Galerkin
solutions of (1), (2) in the form of finite sums
um(x, t) = m∑
Ck(t)ψk(x), (182)
where expansion coefficients {Cj} solve a quadratic dynamical system (DS): C′j =
∑ (k,l) CkClJklj , where Jklj = 〈ψkψ′l, ψ′′j 〉 = λj〈ψkψ′l, ψj〉. (183)
For the conservation law (17), the DS takes the same form as in (183), with the only difference that
Jklj = −〈ψkψ′l, ψj〉. (184) The identity (33) for um takes the form
∑ (k)
2 k(t) = c0m =
∑ (k)
2 k(0), t > 0. (185)
This guarantees global existence of the solutions um(x, t) showing that
Ck(t) do not blow up and exist for all t > 0. (186)
Since ψk are given by sin(λkx) or cos(λkx), a lot of coefficients Jklj vanish. For instance, if u0(x) is odd, we take all the sin functions,
ψk(x) = 1√ L sin
) , with λk = −k2π2L2 , k = 1, 2, ... .
