Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

Abstract

This article is concerned with formulation of three-dimensional thin film model for an antisurfactant solution and hence constructing unique global solution for a two-dimensional Riemann problem for the corresponding reduced hyperbolic form. We develop six geometrically different structures of the solution using generalized characteristic analysis method while relaxing the restriction that only one planar elementary wave is developed at the interface of each initial discontinuity. We analyze the interactions of classical and nonclassical waves in detail to construct the global solution of the corresponding 2-D Riemann problem. Further, we provide the expressions for strength, location, and propagation speed of delta shock wave at each interaction point. Moreover, we compare these solutions with the solutions of a one-dimensional rotated initial value problem and prove that our solutions are globally unique.

Existence of solutions to gas expansion problem through a sharp corner for 2‐D Euler equations with general equation of state

“…Now we move forward to prove the other part of the lemma. Now if 0 < 𝛼 − 𝛽 = 𝜖 2 < 𝛼 0 + 𝜋 2 at some point in Ω τ, then we see from (15) that…”

“…[18] for more details. We use the commutator relation (17) on 𝑢 and use (16) to obtain Then one can apply the commutator relation on 𝜌 and use the relations (15) to prove this proposition. ■ Proposition 3.…”

“…In recent years, a lot of significant work has been done for the 2-D compressible Euler system as well as numerous other related models for a variety of initial and boundary value problems; see, namely, Refs. [5][6][7][8][9][10][11][12][13][14][15][16]. In particular, for gas expansion problems through a sharp corner or wedge, we refer the reader to Refs.…”