On the existence and regularity of solutions of semihyperbolic patches to 2‐D Euler equations with van der Waals gas

Abstract

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.

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

“…Proof. We substitute (𝑢, 𝑣, 𝜏)(𝜉, 𝜂) = ( û, v, τ)(𝜉, 𝛼) and 𝜂 = 𝜉 tan 𝛼 into pseudo-Bernoulli's law (6) and the system (9) to obtain…”

“…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.…”

Existence and regularity of solutions of a supersonic-sonic patch arising in axisymmetric relativistic transonic flow with general equation of state

“…However, unlike the Mach-flow angle plane, the reduced hyperbolic equations in our case do not form a closed system and additional equations are needed to be added to the system in order to close the system which makes the current problem even more complicated. We also comment that the derivation of a priori estimates of solutions for the current problem is also not very easy as a priori estimates developed in the previous works such as [6,7,46,48,49] which are based on characteristic decompositions in homogeneous form. But the nonhomogeneous terms in this problem lead us to the nonhomogeneous form of characteristic decompositions of the angle variables, which greatly affect the establishment of a priori estimates of the solutions.…”

“…One of the main complexity of the problem under consideration is that the velocity data is given only on the streamline arc P E in contrast to all other sonic-supersonic boundary value problems or semi-hyperbolic patch problems where the data is prescribed not only on a streamline but also on a characteristic curve as well (see for example [17,18,45,46]). In particular, for relativistic Euler equations, we refer readers to [47].…”

Degenerate Cauchy–Goursat problem for 2‐D steady isentropic Euler system with van der Waals gas

“…, 𝐴 > 0 depends on the entropy of the system, 𝑎 > 0 represents an attraction energy among the molecules of gas, and 𝑏 > 0 is the covolume of these molecules (see Refs. 38,39). The equation of state, given by Equation (2), can be considered as a perfect gas adulterated by dusty particles for 𝑎 = 0 and a polytropic perfect gas for 𝑎 = 𝑏 = 0 (see Ref.…”