2,909 research outputs found
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-
Generalized Lame operators
We introduce a class of multidimensional Schr\"odinger operators with
elliptic potential which generalize the classical Lam\'e operator to higher
dimensions. One natural example is the Calogero--Moser operator, others are
related to the root systems and their deformations. We conjecture that these
operators are algebraically integrable, which is a proper generalization of the
finite-gap property of the Lam\'e operator. Using earlier results of Braverman,
Etingof and Gaitsgory, we prove this under additional assumption of the usual,
Liouville integrability. In particular, this proves the Chalykh--Veselov
conjecture for the elliptic Calogero--Moser problem for all root systems. We
also establish algebraic integrability in all known two-dimensional cases. A
general procedure for calculating the Bloch eigenfunctions is explained. It is
worked out in detail for two specific examples: one is related to B_2 case,
another one is a certain deformation of the A_2 case. In these two cases we
also obtain similar results for the discrete versions of these problems,
related to the difference operators of Macdonald--Ruijsenaars type.Comment: 38 pages, latex; in the new version a reference was adde
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
Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem
In this work we consider the identifiability of two coefficients and
in a quasilinear elliptic partial differential equation from observation
of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov
[On uniqueness in inverse problems for semilinear parabolic equations. Archive
for Rational Mechanics and Analysis, 1993] and special singular solutions to
first determine and for . Based on this partial
result, we are then able to determine for by an
adjoint approach.Comment: 10 pages; Proof of Theorem 4.1 correcte
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