270 research outputs found
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
Two-Dimensional Steady Supersonic Exothermically Reacting Euler Flow past Lipschitz Bending Walls
We are concerned with the two-dimensional steady supersonic reacting Euler
flow past Lipschitz bending walls that are small perturbations of a convex one,
and establish the existence of global entropy solutions when the total
variation of both the initial data and the slope of the boundary is
sufficiently small. The flow is governed by an ideal polytropic gas and
undergoes a one-step exothermic chemical reaction under the reaction rate
function that is Lipschtiz and has a positive lower bound. The heat released by
the reaction may cause the total variation of the solution to increase along
the flow direction. We employ the modified wave-front tracking scheme to
construct approximate solutions and develop a Glimm-type functional by
incorporating the approximate strong rarefaction waves and Lipschitz bending
walls to obtain the uniform bound on the total variation of the approximate
solutions. Then we employ this bound to prove the convergence of the
approximate solutions to a global entropy solution that contains a strong
rarefaction wave generated by the Lipschitz bending wall. In addition, the
asymptotic behavior of the entropy solution in the flow direction is also
analyzed.Comment: 58 pages, 16 figures; SIAM J. Math. Anal. (accepted on November 1,
2016
Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows past Lipschitz Perturbed Cones
We are concerned with the structural stability of conical shocks in the
three-dimensional steady supersonic flows past Lipschitz perturbed cones whose
vertex angles are less than the critical angle. The flows under consideration
are governed by the steady isothermal Euler equations for potential flow with
axisymmetry so that the equations contain a singular geometric source term. We
first formulate the shock stability problem as an initial-boundary value
problem with the leading conical shock-front as a free boundary, and then
establish the existence and asymptotic behavior of global entropy solutions in
of the problem. To achieve this, we first develop a modified Glimm scheme
to construct approximate solutions via self-similar solutions as building
blocks in order to incorporate with the geometric source term. Then we
introduce the Glimm-type functional, based on the local interaction estimates
between weak waves, the strong leading conical shock, and self-similar
solutions, as well as the estimates of the center changes of the self-similar
solutions. To make sure the decreasing of the Glimm-type functional, we choose
appropriate weights by careful asymptotic analysis of the reflection
coefficients in the interaction estimates, when the Mach number of the incoming
flow is sufficiently large. Finally, we establish the existence of global
entropy solutions involving a strong leading conical shock-front, besides weak
waves, under the conditions that the Mach number of the incoming flow is
sufficiently large and the weighted total variation of the slopes of the
generating curve of the Lipschitz perturbed cone is sufficiently small.
Furthermore, the entropy solution is shown to approach asymptotically the
self-similar solution that is determined by the incoming flow and the
asymptotic tangent of the cone boundary at infinity.Comment: 50 pages; 7 figue
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