2,203 research outputs found
Global Steady Subsonic Flows through Infinitely Long Nozzles for the Full Euler Equations
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 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.Comment: 32 pages, 1 figure. arXiv admin note: text overlap with
arXiv:0907.3276 by other author
Weakly Nonlinear Geometric Optics for Hyperbolic Systems of Conservation Laws
We present a new approach to analyze the validation of weakly nonlinear
geometric optics for entropy solutions of nonlinear hyperbolic systems of
conservation laws whose eigenvalues are allowed to have constant multiplicity
and corresponding characteristic fields to be linearly degenerate. The approach
is based on our careful construction of more accurate auxiliary approximation
to weakly nonlinear geometric optics, the properties of wave front-tracking
approximate solutions, the behavior of solutions to the approximate asymptotic
equations, and the standard semigroup estimates. To illustrate this approach
more clearly, we focus first on the Cauchy problem for the hyperbolic systems
with compact support initial data of small bounded variation and establish that
the estimate between the entropy solution and the geometric optics
expansion function is bounded by , {\it independent of} the
time variable. This implies that the simpler geometric optics expansion
functions can be employed to study the behavior of general entropy solutions to
hyperbolic systems of conservation laws. Finally, we extend the results to the
case with non-compact support initial data of bounded variation.Comment: 30 pages, 2 figure
Stability of Attached Transonic Shocks in Steady Potential Flow past Three-Dimensional Wedges
We develop a new approach and employ it to establish the global existence and
nonlinear structural stability of attached weak transonic shocks in steady
potential flow past three-dimensional wedges; in particular, the restriction
that the perturbation is away from the wedge edge in the previous results is
removed. One of the key ingredients is to identify a "good" direction of the
boundary operator of a boundary condition of the shock along the wedge edge,
based on the non-obliqueness of the boundary condition for the weak shock on
the edge. With the identification of this direction, an additional boundary
condition on the wedge edge can be assigned to make sure that the shock is
attached on the edge and linearly stable under small perturbation. Based on the
linear stability, we introduce an iteration scheme and prove that there exists
a unique fixed point of the iteration scheme, which leads to the global
existence and nonlinear structural stability of the attached weak transonic
shock. This approach is based on neither the hodograph transformation nor the
spectrum analysis, and should be useful for other problems with similar
difficulties.Comment: 28 Pages; 2 figure
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