This article is concerned in establishing the existence and regularity of solution of semihyperbolic patch problem for two‐dimensional isentropic Euler equations with van der Waals gas. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is very common in the numerical solution of Riemann problems. We use the idea of characteristic decomposition and bootstrap method to prove the existence of a global smooth solution, which is uniformly C1,12$C^{1, \tfrac{1}{2}}$ continuous up to the sonic curve. We also prove that the sonic curve is C1,12$C^{1, \tfrac{1}{2}}$ continuous. Further, we show the formation of shock as an envelope for positive characteristics before reaching their sonic points.
In this article, we construct non-self-similar Riemann solutions for a two-dimensional quasilinear hyperbolic system of conservation laws which describes the fluid flow in a thin film for a perfectly soluble anti-surfactant solution. The initial Riemann data consists of two different constant states separated by a smooth curve in
x
−
y
x-y
plane, so without using self-similarity transformation and dimension reduction, we establish solutions for five different cases. Further, we consider interaction of all possible nonlinear waves by taking initial discontinuity curve as a parabola to develop the structure of global entropy solutions explicitly.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.