78 research outputs found
Approximations of pseudo-differential flows
Given a classical symbol of order zero, and associated semiclassical
operators we prove that the flow of is well approximated, in time by a
pseudo-differential operator, the symbol of which is the flow of
the symbol A similar result holds for non-autonomous equations, associated
with time-dependent families of symbols This result was already used,
by the author and co-authors, to give a stability criterion for high-frequency
WKB approximations, and to prove a strong Lax-Mizohata theorem. We give here
two further applications: sharp semigroup bounds, implying nonlinear
instability under the assumption of spectral instability at the symbolic level,
and a new proof of sharp G\r{a}rding inequalities.Comment: Final version, to appear in Indiana Univ. Math.
Nash--Moser iteration and singular perturbations
We present a simple and easy-to-use Nash--Moser iteration theorem tailored
for singular perturbation problems admitting a formal asymptotic expansion or
other family of approximate solutions depending on a parameter \eps\to 0. The
novel feature is to allow loss of powers of \eps as well as the usual loss of
derivatives in the solution operator for the associated linearized problem. We
indicate the utility of this theorem by describing sample applications to (i)
systems of quasilinear Schr\"odinger equations, and (ii) existence of
small-amplitude profiles of quasilinear relaxation systems.Comment: Final version; general presentation completely revised; new sample
application to systems of quasilinear Schr\"odinger equation
A stability criterion for high-frequency oscillations
We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility
condition involves the hyperbolic operator, the fundamental phase associated
with the initial oscillation, and the semilinear source term; it states roughly
that hyperbolicity is preserved around resonances.
If the compatibility condition is satisfied, the solutions are defined over
time intervals independent of the wavelength, and the associated WKB solutions
are stable under a large class of initial perturbations. If the compatibility
condition is not satisfied, resonances are exponentially amplified, and
arbitrarily small initial perturbations can destabilize the WKB solutions in
small time.
The amplification mechanism is based on the observation that in frequency
space, resonances correspond to points of weak hyperbolicity. At such points,
the behavior of the system depends on the lower order terms through the
compatibility condition.
The analysis relies, in the unstable case, on a short-time Duhamel
representation formula for solutions of zeroth-order pseudo-differential
equations.
Our examples include coupled Klein-Gordon systems, and systems describing
Raman and Brillouin instabilities.Comment: Final version, to appear in M\'em. Soc. Math. F
On nonlinear stabilization of linearly unstable maps
We examine the phenomenon of nonlinear stabilization, exhibiting a variety of
related examples and counterexamples. For G\^ateaux differentiable maps, we
discuss a mechanism of nonlinear stabilization, in finite and infinite
dimensions, which applies in particular to hyperbolic partial differential
equations, and, for Fr\'echet differentiable maps with linearized operators
that are normal, we give a sharp criterion for nonlinear exponential
instability at the linear rate. These results highlight the fundamental open
question whether Fr\'echet differentiability is sufficient for linear
exponential instability to imply nonlinear exponential instability, at possibly
slower rate.Comment: New section 1.5 and several references added. 20 pages, no figur
Pointwise Green function bounds and stability of combustion waves
Generalizing similar results for viscous shock and relaxation waves, we
establish sharp pointwise Green function bounds and linearized and nonlinear
stability for traveling wave solutions of an abstract viscous combustion model
including both Majda's model and the full reacting compressible Navier--Stokes
equations with artificial viscosity with general multi-species reaction and
reaction-dependent equation of state, % under the necessary conditions of
strong spectral stability, i.e., stable point spectrum of the linearized
operator about the wave, transversality of the profile as a connection in the
traveling-wave ODE, and hyperbolic stability of the associated Chapman--Jouguet
(square-wave) approximation. Notably, our results apply to combustion waves of
any type: weak or strong, detonations or deflagrations, reducing the study of
stability to verification of a readily numerically checkable Evans function
condition. Together with spectral results of Lyng and Zumbrun, this gives
immediately stability of small-amplitude strong detonations in the small
heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending
previous results obtained by energy methods by Liu--Ying and Tesei--Tan for
Majda's model and the reactive Navier--Stokes equations, respectively
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