Recent Preprints by Gui-Qiang Chen Since 2011

Journal Articles

 1.      Myoungjean Bae, Gui-Qiang G. Chen & Mikhail Feldman. Prandtl-Meyer Reflection for Supersonic Flow past a Solid Ramp. December 2011. pdf-file.    Abstract: We present our recent results on the Prandtl-Meyer reflection for supersonic potential flow past a solid ramp. When a steady supersonic flow passes a solid ramp, there are two possible configurations: the weak shock solution and the strong shock solution. Elling-Liu's theorem (2008) indicates that the steady supersonic weak shock solution can be regarded as a long-time asymptotics of an unsteady flow for a class of physical parameters determined by certain assumptions for potential flow. In this paper we discuss our recent progress in removing these assumptions and establishing the stability theorem for steady supersonic weak shock solutions as the long-time asymptotics of unsteady flows for all the physical parameters for potential flow. We apply new mathematical techniques developed in our recent work to obtain monotonicity properties and uniform apriori estimates for weak solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for the general case. 2.      Gui-Qiang G. Chen & James Glimm. Kolmogorov's Theory of Turbulence and Inviscid Limit of the Navier-Stokes Equations in $\R^3$,  November 2011. pdf-file.  Communications in Mathematical Physics (to appear 2012)   Abstract: We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations in $\R^3$. We first observe that a pathwise Kolmogorov hypothesis implies the uniform boundedness of the $\alpha^{th}$-order fractional derivatives of the velocity for some $\alpha>0$ in the space variables in $L^2$, which is independent of the viscosity $\mu>0$. Then it is shown that this key observation yields the $L^2$-equicontinuity in the time variable and the uniform bound in $L^q$, for some $q>2$, of the velocity independent of $\mu>0$. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations in $\R^3$. We also consider passive scalars coupled to the incompressible Navier-Stokes equations and, in this case, find the weak-star convergence for the passive scalars with a limit in the form of a Young measure (pdf depending on space and time). Not only do we offer a framework for mathematical existence theories, but also we offer a framework for the interpretation of numerical solutions through the identification of a function space in which convergence should take place, with the bounds that are independent of $\mu>0$, that is in the high Reynolds number limit. 3.      Gui-Qiang G. Chen, Xuemei Deng & Wei Xiang.   Abstract: We are concerned with global steady subsonic flows through general infinitely long nozzles for the full Euler equations. The problem is formulated as a boundary value problem in the unbounded domain for a nonlinear elliptic equation of second order in terms of the stream function. It is established that, when the oscillation of the entropy and Bernoulli functions at the upstream is sufficiently small in $C^{1,1}$ and the mass flux is in a suitable regime, there exists a unique global subsonic solution in a suitable class of general nozzles. The assumptions are required to prevent from the occurrence of supersonic bubbles inside the nozzles. The asymptotic behavior of subsonic flows at the downstream and upstream, as well as the critical mass flux, have been clarified. 4.      Gui-Qiang G. Chen, Qiang Ding & Kenneth Karlsen. On Nonlinear Stochastic Balance Laws. November 2011. pdf-file.   Archive for Rational Mechanics and Analysis (in press, 2012)   Abstract: We are concerned with multidimensional stochastic balance laws. We identify a class of nonlinear balance laws for which uniform spatial $BV$ bounds for vanishing viscosity approximations can be achieved. Moreover, we establish temporal equicontinuity in $L^1$ of the approximations, uniformly in the viscosity coefficient. Using these estimates, we supply a multidimensional existence theory of stochastic entropy solutions. In addition, we establish an error estimate for the stochastic viscosity method, as well as an explicit estimate for the continuous dependence of stochastic entropy solutions on the flux and random source functions. Various further generalizations of the results are discussed. 5.      Gui-Qiang G. Chen & Hairong Yuan.   Abstract: We establish the local uniqueness of steady transonic shock solutions with spherical symmetry for the three-dimensional full Euler equations. These transonic shock-fronts are important for understanding transonic shock phenomena in divergent nozzles. From mathematical point of view, we show the uniqueness of solutions of a free boundary problem for a multidimensional quasilinear system of mixed-composite elliptic--hyperbolic type. To this end, we develop a decomposition of the Euler system which works in a general Riemannian manifold, a method to study a Venttsel problem of nonclassical nonlocal elliptic operators, and an iteration mapping which possesses locally a unique fixed point. The approach reveals an intrinsic structure of the steady Euler system and subtle interactions of its elliptic and hyperbolic part.

Some of Publications by Gui-Qiang Chen before 2009

Books:

• 1998: Advances in Nonlinear Partial Differential Equations and Related Areas, World Scientific Publishing Co Pte Ltd, ISBN: 981-02-3664-6 (Edited with Y.-Y. Li, X.-P. Zhu, and D.-M. Cao).
• 1999: Nonlinear Partial Differential Equations. Contemporary Mathematics, 238, AMS: Providence, ISBN 0-8218-1196-7 (Edited with E. DiBenedetto).
• 2005: Nonlinear Partial Differential Equations and Related Analysis. Contemporary Mathematics, 371, AMS: Providence, ISBN 0-8218-3533-5 (Edited with G. Gasper and J. Jerome).
• 2007: Stochastic Analysis and Partial Differential Equations, Contemporary Mathematics, 422, AMS: Providence, ISBN-10: 0-8218-4059-2, ISBN-13: 978-0-8218-4059-7 (Edited with E. Hsu and M. Pinsky).
• 2009: Nonlinear Conservation Laws, Fluid Systems and Related Topics, Higher Education Press and World Scientific: Beijing and Singapore, 2009, SBN 7-04-017358-1; ISBN 978-981-4273-27-5, 981-4273-27-9(US) (Edited with Tatsien Li and Chun Liu).

Lecture Notes:

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