# American Institute of Mathematical Sciences

December  2019, 12(6): 1197-1228. doi: 10.3934/krm.2019046

## Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

 1 Department of Mathematics, Hangzhou Normal University, Hangzhou, 310036, China 2 Department of Mathematics, University of Iowa, Iowa City, IA 52242, United States

* Corresponding author: Yanbo Hu

Received  June 2018 Revised  May 2019 Published  September 2019

Fund Project: The first author is supported by NSF of Zhejiang Province of China grant LY17A010019, NSFC grants 11301128, 11571088 and China Scholarship Council grant 201708330155

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

Citation: Yanbo Hu, Tong Li. Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations. Kinetic & Related Models, 2019, 12 (6) : 1197-1228. doi: 10.3934/krm.2019046
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##### References:
 [1] Yuxi Zheng. Absorption of characteristics by sonic curve of the two-dimensional Euler equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 605-616. doi: 10.3934/dcds.2009.23.605 [2] Andrew Raich. Heat equations and the Weighted $\bar\partial$-problem. Communications on Pure & Applied Analysis, 2012, 11 (3) : 885-909. doi: 10.3934/cpaa.2012.11.885 [3] Gui-Qiang Chen, Jun Chen, Mikhail Feldman. Transonic flows with shocks past curved wedges for the full Euler equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (8) : 4179-4211. doi: 10.3934/dcds.2016.36.4179 [4] Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775 [5] Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 [6] Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837 [7] Kyeong-Hun Kim, Kijung Lee. A weighted $L_p$-theory for second-order parabolic and elliptic partial differential systems on a half space. Communications on Pure & Applied Analysis, 2016, 15 (3) : 761-794. doi: 10.3934/cpaa.2016.15.761 [8] Gui-Qiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shock-fronts for the three--dimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2515-2542. doi: 10.3934/cpaa.2013.12.2515 [9] Guanghui Hu, Yavar Kian. Determination of singular time-dependent coefficients for wave equations from full and partial data. Inverse Problems & Imaging, 2018, 12 (3) : 745-772. doi: 10.3934/ipi.2018032 [10] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [11] Igor Kukavica, Vlad C. Vicol. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 285-303. doi: 10.3934/dcds.2011.29.285 [12] Heiko Enderling, Alexander R.A. Anderson, Mark A.J. Chaplain, Glenn W.A. Rowe. Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology. Mathematical Biosciences & Engineering, 2006, 3 (4) : 571-582. doi: 10.3934/mbe.2006.3.571 [13] Ansgar Jüngel, Josipa-Pina Milišić. Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution. Kinetic & Related Models, 2011, 4 (3) : 785-807. doi: 10.3934/krm.2011.4.785 [14] Linglong Du. Characteristic half space problem for the Broadwell model. Networks & Heterogeneous Media, 2014, 9 (1) : 97-110. doi: 10.3934/nhm.2014.9.97 [15] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [16] Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141 [17] Carolyn Mayer, Kathryn Haymaker, Christine A. Kelley. Channel decomposition for multilevel codes over multilevel and partial erasure channels. Advances in Mathematics of Communications, 2018, 12 (1) : 151-168. doi: 10.3934/amc.2018010 [18] Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733 [19] Norisuke Ioku. Some space-time integrability estimates of the solution for heat equations in two dimensions. Conference Publications, 2011, 2011 (Special) : 707-716. doi: 10.3934/proc.2011.2011.707 [20] Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45

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