13,633 research outputs found
Well-posedness and qualitative behaviour of a semi-linear parabolic Cauchy problem arising from a generic model for fractional-order autocatalysis
In this paper, we examine a semi-linear parabolic Cauchy problem with non-Lipschitz nonlinearity which arises as a generic form in a significant number of applications. Specifically, we obtain a well-posedness result and examine the qualitative structure of the solution in detail. The standard classical approach to establishing well-posedness is precluded owing to the lack of Lipschitz continuity for the nonlinearity. Here, existence and uniqueness of solutions is established via the recently developed generic approach to this class of problem (Meyer & Needham 2015 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations. London Mathematical Society Lecture Note Series, vol. 419) which examines the difference of the maximal and minimal solutions to the problem. From this uniqueness result, the approach of Meyer & Needham allows for development of a comparison result which is then used to exhibit global continuous dependence of solutions to the problem on a suitable initial dataset. The comparison and continuous dependence results obtained here are novel to this class of problem. This class of problem arises specifically in the study of a one-step autocatalytic reaction, which is schematically given by A→B at rate a(p)b(q) (where a and b are the concentrations of A and B, respectively, with 0<p,q<1) and well-posedness for this problem has been lacking up to the present
Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces
For both localized and periodic initial data, we prove local existence in
classical energy space , for a class of dispersive
equations with nonlinearities of mild regularity.
Our results are valid for symmetric Fourier multiplier operators whose
symbol is of temperate growth, and in local Sobolev space
. In particular, the results include
non-smooth and exponentially growing nonlinearities. Our proof is based on a
combination of semi-group methods and a new composition result for Besov
spaces. In particular, we extend a previous result for Nemytskii operators on
Besov spaces on to the periodic setting by using the
difference-derivative characterization of Besov spaces
Outer boundary conditions for Einstein's field equations in harmonic coordinates
We analyze Einstein's vacuum field equations in generalized harmonic coordinates on a compact spatial domain with boundaries. We specify a class of boundary conditions, which is constraint-preserving and sufficiently general to include recent proposals for reducing the amount of spurious reflections of gravitational radiation. In particular, our class comprises the boundary conditions recently proposed by Kreiss and Winicour, a geometric modification thereof, the freezing-Ψ0 boundary condition and the hierarchy of absorbing boundary conditions introduced by Buchman and Sarbach. Using the recent technique developed by Kreiss and Winicour based on an appropriate reduction to a pseudo-differential first-order system, we prove well posedness of the resulting initial-boundary value problem in the frozen coefficient approximation. In view of the theory of pseudo-differential operators, it is expected that the full nonlinear problem is also well posed. Furthermore, we implement some of our boundary conditions numerically and study their effectiveness in a test problem consisting of a perturbed Schwarzschild black hole
Mixed Hyperbolic - Second-Order Parabolic Formulations of General Relativity
Two new formulations of general relativity are introduced. The first one is a
parabolization of the Arnowitt, Deser, Misner (ADM) formulation and is derived
by addition of combinations of the constraints and their derivatives to the
right-hand-side of the ADM evolution equations. The desirable property of this
modification is that it turns the surface of constraints into a local attractor
because the constraint propagation equations become second-order parabolic
independently of the gauge conditions employed. This system may be classified
as mixed hyperbolic - second-order parabolic. The second formulation is a
parabolization of the Kidder, Scheel, Teukolsky formulation and is a manifestly
mixed strongly hyperbolic - second-order parabolic set of equations, bearing
thus resemblance to the compressible Navier-Stokes equations. As a first test,
a stability analysis of flat space is carried out and it is shown that the
first modification exponentially damps and smoothes all constraint violating
modes. These systems provide a new basis for constructing schemes for long-term
and stable numerical integration of the Einstein field equations.Comment: 19 pages, two column, references added, two proofs of well-posedness
added, content changed to agree with submitted version to PR
Local ill-posedness of the 1D Zakharov system
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov
system for any dimension , in the inhomogeneous Sobolev spaces for a range of exponents ,
depending on . Here we restrict to dimension and present a few results
establishing local ill-posedness for exponent pairs outside of the
well-posedness regime. The techniques employed are rooted in the work of
Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and
Christ-Colliander-Tao (2003) applied to the nonlinear Schroedinger equation
Nonlinear wave equations
The analysis of nonlinear wave equations has experienced a dramatic growth in
the last ten years or so. The key factor in this has been the transition from
linear analysis, first to the study of bilinear and multilinear wave
interactions, useful in the analysis of semilinear equations, and next to the
study of nonlinear wave interactions, arising in fully nonlinear equations. The
dispersion phenomena plays a crucial role in these problems. The purpose of
this article is to highlight a few recent ideas and results, as well as to
present some open problems and possible future directions in this field
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