" Construction of the general evolution equation of pseuduvector solenoidal field with local conservation law "

Description of a Class of Evolutionary Equations in Ferrodynamics

“…here the left-hand side contains the decomposition over linearly independent invariants consisting of the tensors A jk and B jkl and the pseudovector M j . The equation obtained, which holds for any solenoidal, unimodal, twice continuously differentiable field M on R 3 , implies f (16) = 0, f (14) = 0, f (12) = f (17) .…”

“…A simple algebraic analysis leads us to the following assertion. Theorem 4.1 (see [14]). The set U is empty.…”

“…If, in addition, the field M (x, t) is solenoidal, i.e., (∇, M ) = 0, then the flux densities with numbers 1, 9, and 15 also vanish due to the divergence (∇, M ). As a result, under the unimodality and solenoidality conditions, the general equation Ṁj = L j [M ], which governs the dynamics of the field M and contains the differential operator described in Theorem 4.2 and the formula (4.1), takes the form Ṁj = f (2) − f (3) ΔM j + f (7) + f (8) ∇ k M j (M , ∇)M k + f (7) − f (8) ∇ k M k (M , ∇)M j + f (10) ∇ j M , [∇, M ] + f (11) + f (12) ∇ k [M , ∇] j M k + f (11) − f (12) ∇ k [M , ∇] k M j + ε jln f (13) + f (14) ∇ k (M n ∇ k M l ) + ε kln f (13) − f (14)…”

“…I. The first equality (6.2) yields the following equation for the coefficients f (α) : f (7) (10) Δ M , [∇, M ] + f (11) (6.3) whereas the second to the equation f (2) − f (3) M j ΔM j + f (10) M j ∇ j M , [∇, M ] + ε kln f (13) − f (14)…”

“…(6.15) Substituting m = A exp(x, k) into Eqs. (6.12)- (6.15), where the vectors k and A are such that (A, M (0) ) = 0 and (A, k) = 0, respectively, we obtain in each equality nonzero factors depending on the field m. Then f (10) + 2f (13) = 0, f (20) = 2f (22) , f (22) = f (23) , (6.16) f (18) = f (13) + f (14) + f (17) − f (16) . (6.17)…”