Weak asymptotics method and interaction of nonlinear waves

Approximate solutions to the initial value problem for some compressible flows

“…An explanation is needed to justify the convolution in (18). The formulas (15,16) state that the conservation laws of mass and momentum are valid in the cells of length . These conservation laws are known to hold at a very small scale close to the molecular scale; maybe can be of the order of magnitude of 10 −8 m. On the other hand, the state law p = Kρ is valid only at the order of magnitude of the measurement apparatus and in absence of strong irregularities such as shock waves, or a slight extrapolation at smaller order, maybe at a magnitude not smaller than 10 −3 m represented in (18) by α in the convolution.…”

“…The solution of the system of ODEs (15)(16)(17)(18), with α < 1 6 , 3α + β < N − 1, with initial conditions (ρ 0 , u 0 ), provides a weak asymptotic solution (1) for the 1-D isothermal gas equations, which is global in time t ∈ [0, +∞[ and in space x ∈ T. When α < 1 4+2n and (2 + n)α + β < N − 1, the result holds as well in n-D, n=2,3 for a direct extension of (15,18).…”

“…The need for the search of mathematical solutions to the initial value problem for compressible fluids in several dimensions when shocks show up has been recently pointed out by Lax and Serre [25,31]; our results provide sequences of approximate solutions with full mathematical proofs that tend to satisfy the equations. We have been motivated by the fact our approximate solutions are weak asymptotic solutions as considered by Albeverio et al [1][2][3], Danilov et al [15][16][17][18], Shelkovich et al [30,32,33], as an extension of the Maslov-Whitham asymptotic analysis. Exact or approximate solutions with full proofs in the case of spherical symmetry had already been obtained in Joseph et al [4,22] and Kunzinger et al [23,24] for some of these systems or related systems.…”

“…Colombeau ZAMP tool in itself by various authors following Maslov asymptotic analysis [1][2][3][15][16][17][18]30,32,33], and also, they could be a provisional substitute of solutions, since one has to cope with an absence of known usual weak solutions in 2-D and 3-D in the presence of shocks. The proof that these sequences tend to satisfy the equations permits to explain results observed from computing in [9,25,26].…”

“…These sequences of approximate solutions have been introduced under the name of weak asymptotic solutions by Danilov et al [15], as an extension of Maslov asymptotic analysis, and they have proved to be an efficient mathematical tool to study creation and superposition of singular solutions to various nonlinear PDEs, in particular δ-waves: [1][2][3][15][16][17][18]30,32,33]. A weak asymptotic solution for the system…”