Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure

Loss of Regularity of Solutions of the Lighthill Problem for Shock Diffraction for Potential Flow

We are concerned with the suitability of the main models of compressible fluid dynamics for the Lighthill problem for shock diffraction by a convex corned wedge, by studying the regularity of solutions of the problem, which can be formulated as a free boundary problem. In this paper, we prove that there is no regular solution that is subsonic up to the wedge corner for potential flow. This indicates that, if the solution is subsonic at the wedge corner, at least a characteristic discontinuity (vortex sheet or entropy wave) is expected to be generated, which is consistent with the experimental and computational results. Therefore, the potential flow equation is not suitable for the Lighthill problem so that the compressible Euler system must be considered. In order to achieve the non-existence result, a weak maximum principle for the solution is established, and several other mathematical techniques are developed. The methods and techniques developed here are also useful to the other problems with similar difficulties.Comment: 20 pages, 4 figures, To appear in: SIAM Journal of Mathematical Analysis, 202

Convexity of Self-Similar Transonic Shocks and Free Boundaries for the Euler Equations for Potential Flow

We are concerned with geometric properties of transonic shocks as free boundaries in two-dimensional self-similar coordinates for compressible fluid flows, which are not only important for the understanding of geometric structure and stability of fluid motions in continuum mechanics but also fundamental in the mathematical theory of multidimensional conservation laws. A transonic shock for the Euler equations for self-similar potential flow separates elliptic (subsonic) and hyperbolic (supersonic) phases of the self-similar solution of the corresponding nonlinear partial differential equation in a domain under consideration, in which the location of the transonic shock is apriori unknown. We first develop a general framework under which self-similar transonic shocks, as free boundaries, are proved to be uniformly convex, and then apply this framework to prove the uniform convexity of transonic shocks in the two longstanding fundamental shock problems -- the shock reflection-diffraction by wedges and the Prandtl-Meyer reflection for supersonic flows past solid ramps. To achieve this, our approach is to exploit underlying nonlocal properties of the solution and the free boundary for the potential flow equation.Comment: 81 pages; 12 figures; To appear in Archive for Rational Mechanics and Analysis (2020

NASA Thesaurus. Volume 2: Access vocabulary

The NASA Thesaurus -- Volume 2, Access Vocabulary -- contains an alphabetical listing of all Thesaurus terms (postable and nonpostable) and permutations of all multiword and pseudo-multiword terms. Also included are Other Words (non-Thesaurus terms) consisting of abbreviations, chemical symbols, etc. The permutations and Other Words provide 'access' to the appropriate postable entries in the Thesaurus

The Morawetz Problem for Supersonic Flow with Cavitation

We are concerned with the existence and compactness of entropy solutions of the compressible Euler system for two-dimensional steady potential flow around an obstacle for a polytropic gas with supersonic far-field velocity. The existence problem, initially posed by Morawetz \cite{morawetz85} in 1985, has remained open since then. In this paper, we establish the first complete existence theorem for the Morawetz problem by developing a new entropy analysis, coupled with a vanishing viscosity method and compensated compactness ideas. The main challenge arises when the flow approaches cavitation, leading to a loss of strict hyperbolicity of the system and a singularity of the entropy equation, particularly for the case of adiabatic exponent $\gamma=3$. Our analysis provides a complete description of the entropy and entropy-flux pairs via the Loewner--Morawetz relations, which, in turn, leads to the establishment of a compensated compactness framework. As direct applications of our entropy analysis and the compensated compactness framework, we obtain the compactness of entropy solutions and the weak continuity of the compressible Euler system in the supersonic regime.Comment: 59 pages, 2 figure

NASA thesaurus. Volume 2: Access vocabulary

The Access Vocabulary, which is essentially a permuted index, provides access to any word or number in authorized postable and nonpostable terms. Additional entries include postable and nonpostable terms, other word entries, and pseudo-multiword terms that are permutations of words that contain words within words. The Access Vocabulary contains 40,738 entries that give increased access to the hierarchies in Volume 1 - Hierarchical Listing

NASA thesaurus. Volume 1: Hierarchical Listing

There are over 17,000 postable terms and nearly 4,000 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary and Volume 3 - Definitions

NASA Thesaurus. Volume 1: Hierarchical listing

There are 16,713 postable terms and 3,716 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary

NASA thesaurus. Volume 2: Access vocabulary

The access vocabulary, which is essentially a permuted index, provides access to any word or number in authorized postable and nonpostable terms. Additional entries include postable and nonpostable terms, other word entries and pseudo-multiword terms that are permutations of words that contain words within words. The access vocabulary contains almost 42,000 entries that give increased access to the hierarchies in Volume 1 - Hierarchical Listing

NASA thesaurus aeronautics vocabulary

The controlled vocabulary used by the NASA Scientific and Technical Information effort to index documents in the area of aeronautics is presented. The terms comprise a subset of the 1988 edition of the NASA Thesaurus and its supplements issued through the end of 1990. The Aeronautics Vocabulary contains over 4700 terms presented in a hierarchical display format. In addition to aeronautics per se, the vocabulary covers supporting terminology from areas such as fluid dynamics, propulsion engineering, and test facilities and instrumentation