Supersonic Flow onto Solid Wedges, Multidimensional Shock Waves and Free Boundary Problems

When an upstream steady uniform supersonic flow impinges onto a symmetric straight-sided wedge, governed by the Euler equations, there are two possible steady oblique shock configurations if the wedge angle is less than the detachment angle -- the steady weak shock with supersonic or subsonic downstream flow (determined by the wedge angle that is less or larger than the sonic angle) and the steady strong shock with subsonic downstream flow, both of which satisfy the entropy condition. The fundamental issue -- whether one or both of the steady weak and strong shocks are physically admissible solutions -- has been vigorously debated over the past eight decades. In this paper, we survey some recent developments on the stability analysis of the steady shock solutions in both the steady and dynamic regimes. For the static stability, we first show how the stability problem can be formulated as an initial-boundary value type problem and then reformulate it into a free boundary problem when the perturbation of both the upstream steady supersonic flow and the wedge boundary are suitably regular and small, and we finally present some recent results on the static stability of the steady supersonic and transonic shocks. For the dynamic stability for potential flow, we first show how the stability problem can be formulated as an initial-boundary value problem and then use the self-similarity of the problem to reduce it into a boundary value problem and further reformulate it into a free boundary problem, and we finally survey some recent developments in solving this free boundary problem for the existence of the Prandtl-Meyer configurations that tend to the steady weak supersonic or transonic oblique shock solutions as time goes to infinity. Some further developments and mathematical challenges in this direction are also discussed.Comment: 19 pages; 8 figures; accepted by Science China Mathematics on February 22, 2017 (invited survey paper). doi: 10.1007/s11425-016-9045-

Weak Continuity and Compactness for Nonlinear Partial Differential Equations

We present several examples of fundamental problems involving weak continuity and compactness for nonlinear partial differential equations, in which compensated compactness and related ideas have played a significant role. We first focus on the compactness and convergence of vanishing viscosity solutions for nonlinear hyperbolic conservation laws, including the inviscid limit from the Navier-Stokes equations to the Euler equations for homentropy flow, the vanishing viscosity method to construct the global spherically symmetric solutions to the multidimensional compressible Euler equations, and the sonic-subsonic limit of solutions of the full Euler equations for multidimensional steady compressible fluids. We then analyze the weak continuity and rigidity of the Gauss-Codazzi-Ricci system and corresponding isometric embeddings in differential geometry. Further references are also provided for some recent developments on the weak continuity and compactness for nonlinear partial differential equations.Comment: 29 page

Stability of Steady Multi-Wave Configurations for the Full Euler Equations of Compressible Fluid Flow

We are concerned with the stability of steady multi-wave configurations for the full Euler equations of compressible fluid flow. In this paper, we focus on the stability of steady four-wave configurations that are the solutions of the Riemann problem in the flow direction, consisting of two shocks, one vortex sheet, and one entropy wave, which is one of the core multi-wave configurations for the two-dimensional Euler equations. It is proved that such steady four-wave configurations in supersonic flow are stable in structure globally, even under the BV perturbation of the incoming flow in the flow direction. In order to achieve this, we first formulate the problem as the Cauchy problem (initial value problem) in the flow direction, and then develop a modified Glimm difference scheme and identify a Glimm-type functional to obtain the required BV estimates by tracing the interactions not only between the strong shocks and weak waves, but also between the strong vortex sheet/entropy wave and weak waves. The key feature of the Euler equations is that the reflection coefficient is always less than 1, when a weak wave of different family interacts with the strong vortex sheet/entropy wave or the shock wave, which is crucial to guarantee that the Glimm functional is decreasing. Then these estimates are employed to establish the convergence of the approximate solutions to a global entropy solution, close to the background solution of steady four-wave configuration.Comment: 9 figures

Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of compensated compactness. We then deduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: For a family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on a semi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformly bounded in $L^p$ and satisfies the Cartan structural system, then any weak $L^p$ limit of $\{\mathcal{W}_\varepsilon\}$ is also a solution of the Cartan structural system. Moreover, it is proved that isometric immersions of semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci system (Theorem 5.1), which leads to the $L^p$ weak continuity of the Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further applications, the weak continuity of Einstein's constraint equations, general immersed hypersurfaces, and the quasilinear wave equations is also established.Comment: 64 page

Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data

We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental question is whether concentration could form at the origin. In this paper, we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions, and establish the convergence of the approximate solutions to a global finite-energy entropy solution of the compressible Euler equations with spherical symmetry and large initial data. This indicates that concentration does not form in the vanishing viscosity limit, even though the density may blow up at certain time. To achieve this, we first construct global smooth solutions of appropriate initial-boundary value problems for the Euler equations with designed viscosity terms, an approximate pressure function, and boundary conditions, and then we establish the strong convergence of the viscosity approximate solutions to a finite-energy entropy solutions of the Euler equations.Comment: 29 page

Kolmogorov-Type Theory of Compressible Turbulence and Inviscid Limit of the Navier-Stokes Equations in R3\mathbb{R}^3

We are concerned with the inviscid limit of the Navier-Stokes equations to the Euler equations for compressible fluids in $\mathbb{R}^3$. Motivated by the Kolmogorov hypothesis (1941) for incompressible flow, we introduce a Kolmogorov-type hypothesis for barotropic flows, in which the density and the sonic speed normally vary significantly. We then observe that the compressible Kolmogorov-type hypothesis implies the uniform boundedness of some fractional derivatives of the weighted velocity and sonic speed in the space variables in $L^2$, which is independent of the viscosity coefficient $\mu>0$. It is shown that this key observation yields the equicontinuity in both space and time of the density in $L^\gamma$ and the momentum in $L^2$, as well as the uniform bound of the density in $L^{q_1}$ and the velocity in $L^{q_2}$ independent of $\mu>0$, for some fixed $q_1 >\gamma$ and $q_2 >2$, where $\gamma>1$ is the adiabatic exponent. These results lead to the strong convergence of solutions of the Navier-Stokes equations to a solution of the Euler equations for barotropic fluids in $\mathbb{R}^3$. 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.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1008.154