277 research outputs found
Universal dynamics for the defocusing logarithmic Schrodinger equation
We consider the nonlinear Schrodinger equation with a logarithmic
nonlinearity in a dispersive regime. We show that the presence of the
nonlinearity affects the large time behavior of the solution: the dispersion is
faster than usual by a logarithmic factor in time and the positive Sobolev
norms of the solution grow logarithmically in time. Moreover, after rescaling
in space by the dispersion rate, the modulus of the solution converges to a
universal Gaussian profile. These properties are suggested by explicit
computations in the case of Gaussian initial data, and remain when an extra
power-like nonlinearity is present in the equation. One of the key steps of the
proof consists in using the Madelung transform to reduce the equation to a
variant of the isothermal compressible Euler equation, whose large time
behavior turns out to be governed by a parabolic equation involving a
Fokker-Planck operator.Comment: Final versio
Besov algebras on Lie groups of polynomial growth
We prove an algebra property under pointwise multiplication for Besov spaces
defined on Lie groups of polynomial growth. When the setting is restricted to
the case of H-type groups, this algebra property is generalized to paraproduct
estimates
On the stability in weak topology of the set of global solutions to the Navier-Stokes equations
Let be a suitable function space and let \cG \subset X be the set of
divergence free vector fields generating a global, smooth solution to the
incompressible, homogeneous three dimensional Navier-Stokes equations. We prove
that a sequence of divergence free vector fields converging in the sense of
distributions to an element of \cG belongs to \cG if is large enough,
provided the convergence holds "anisotropically" in frequency space. Typically
that excludes self-similar type convergence. Anisotropy appears as an important
qualitative feature in the analysis of the Navier-Stokes equations; it is also
shown that initial data which does not belong to \cG (hence which produces a
solution blowing up in finite time) cannot have a strong anisotropy in its
frequency support.Comment: To appear in Archive for Rational and Mechanical Analysi
Large, global solutions to the Navier-Stokes equations, slowly varying in one direction
In to previous papers by the authors, classes of initial data to the three
dimensional, incompressible Navier-Stokes equations were presented, generating
a global smooth solution although the norm of the initial data may be chosen
arbitrarily large. The aim of this article is to provide new examples of
arbitrarily large initial data giving rise to global solutions, in the whole
space. Contrary to the previous examples, the initial data has no particular
oscillatory properties, but varies slowly in one direction. The proof uses the
special structure of the nonlinear term of the equation.Comment: References adde
Weak convergence results for inhomogeneous rotating fluid equations
We consider the equations governing incompressible, viscous fluids in three
space dimensions, rotating around an inhomogeneous vector B(x): this is a
generalization of the usual rotating fluid model (where B is constant). We
prove the weak convergence of Leray--type solutions towards a vector field
which satisfies the usual 2D Navier--Stokes equation in the regions of space
where B is constant, with Dirichlet boundary conditions, and a heat--type
equation elsewhere. The method of proof uses weak compactness arguments
On the global wellposedness of the 3-D Navier-Stokes equations with large initial data
We give a condition for the periodic, three dimensional, incompressible
Navier-Stokes equations to be globally wellposed. This condition is not a
smallness condition on the initial data, as the data is allowed to be
arbitrarily large in the scale invariant space ,
which contains all the known spaces in which there is a global solution for
small data. The smallness condition is rather a nonlinear type condition on the
initial data; an explicit example of such initial data is constructed, which is
arbitrarily large and yet gives rise to a global, smooth solution
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