153 research outputs found
Dynamics of nonlinear wave equations
In this lecture, we will survey the study of dynamics of the nonlinear wave
equation in recent years. We refer to some lecture notes including such as C.
Kenig \cite{Kenig01,Kenig02}, C. Kenig and F. Merle \cite{KM1} J. Shatah and M.
Struwe \cite{SS98}, and C. Sogge \cite{sogge:wave} etc. This lecture was
written for LIASFMA School and Workshop on Harmonic Analysis and Wave Equations
in Fudan universty.Comment: 80pages. This lecture was written for LIASFMA School on Harmonic
Analysis and Wave Equations in Fudan universty(2017). arXiv admin note: text
overlap with arXiv:1506.00788, arXiv:0710.5934, arXiv:1407.4525,
arXiv:1601.01871, arXiv:1301.4835, 1509.03331, arXiv:1010.3799,
arXiv:1201.3258, arXiv:0911.4534, arXiv:1411.7905, by other author
Scattering theory for energy-supercritical Klein-Gordon equation
In this paper, we consider the question of the global well-posedness and
scattering for the cubic Klein-Gordon equation in
dimension . We show that if the solution is apriorily bounded in
the critical Sobolev space, that is, with , then
is global and scatters. The impetus to consider this problem stems from a
series of recent works for the energy-supercritical nonlinear wave equation and
nonlinear Schr\"odinger equation. However, the scaling invariance is broken in
the Klein-Gordon equation. We will utilize the concentration compactness ideas
to show that the proof of the global well-posedness and scattering is reduced
to disprove the existence of the scenario: soliton-like solutions. And such
solutions are precluded by making use of the Morawetz inequality, finite speed
of propagation and concentration of potential energy.Comment: 24page
Global well-posedness for the two-dimensional Maxwell-Navier-Stokes equations
In this paper, we investigate Cauchy problem of the two-dimensional full
Maxwell-Navier-Stokes system, and prove the global-in-time existence and
uniqueness of solution in the borderline space which is very close to
-energy space by developing the new estimate of . This solves the open
problem in the framework of borderline space purposed by Masmoudi in
\cite{Masmoudi-10}.Comment: 46page
Scattering theory for the defocusing fourth-order Schr\"odinger equation
In this paper, we study the global well-posedness and scattering theory for
the defocusing fourth-order nonlinear Schr\"odinger equation (FNLS)
in dimension . We prove that if the solution
is apriorily bounded in the critical Sobolev space, that is, with all if
is an even integer or otherwise, then is global and
scatters. The impetus to consider this problem stems from a series of recent
works for the energy-supercritical and energy-subcritical nonlinear
Schr\"odinger equation (NLS) and nonlinear wave equation (NLW). We will give a
uniform way to treat the energy-subcritical, energy-critical and
energy-supercritical FNLS, where we utilize the strategy derived from
concentration compactness ideas to show that the proof of the global
well-posedness and scattering is reduced to exclude the existence of three
scenarios: finite time blowup; soliton-like solution and low to high frequency
cascade. Making use of the No-waste Duhamel formula, we deduce that the energy
or mass of the finite time blow-up solution is zero and so get a contradiction.
Finally, we adopt the double Duhamel trick, the interaction Morawetz estimate
and interpolation to kill the last two scenarios.Comment: 40pages. arXiv admin note: text overlap with arXiv:0812.2084 by other
author
Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity
In this paper, we are concerned with the tridimensional anisotropic
Boussinesq equations which can be described by {equation*}
{{array}{ll}
(\partial_{t}+u\cdot\nabla)u-\kappa\Delta_{h} u+\nabla \Pi=\rho
e_{3},\quad(t,x)\in\mathbb{R}^{+}\times\mathbb{R}^{3},
(\partial_{t}+u\cdot\nabla)\rho=0,
\text{div}u=0.
{array}. {equation*} Under the assumption that the support of the
axisymmetric initial data does not intersect the axis ,
we prove the global well-posedness for this system with axisymmetric initial
data. We first show the growth of the quantity for large time by
taking advantage of characteristic of transport equation. This growing property
together with the horizontal smoothing effect enables us to establish
-estimate of the velocity via the -energy estimate of velocity and
the Maximum principle of density. Based on this, we further establish the
estimate for the quantity \|\omega(t)\|_{\sqrt{\mathbb{L}}}:=\sup_{2\leq
p<\infty}\frac{\norm{\omega(t)}_{L^p(\mathbb{R}^3)}}{\sqrt{p}}<\infty which
implies \|\nabla u(t)\|_{\mathbb{L}^{3/2}}:=\sup_{2\leq
p<\infty}\frac{\norm{\nabla u(t)}_{L^p(\mathbb{R}^3)}}{p\sqrt{p}}<\infty.
However, this regularity for the flow admits forbidden singularity since (see \eqref{eq-kl} for the definition) seems be the minimum space
for the gradient vector field ensuring uniqueness of flow. To bridge
this gap, we exploit the space-time estimate about by making good
use of the horizontal smoothing effect and micro-local techniques. The global
well-posedness for the large initial data is achieved by establishing a new
type space-time logarithmic inequality.Comment: 32pages. arXiv admin note: text overlap with arXiv:0908.0894 by other
author
Scattering theory for subcritical wave equation with inverse square potential
We consider the scattering theory for the defocusing energy subcritical wave
equations with an inverse square potential. By employing the energy flux method
of [37], [38] and [41], we establish energy flux estimates on the light cone.
Then by the characteristic line method and radiation theorem, we show the
radial energy solutions scatter to free waves outside the light cone. Moreover,
by extending the Morawetz estimates, we prove scattering theory for the
solutions with initial datum in the weighted energy spaces.Comment: Minor change
On the blow up phenomenon for the -critical focusing Hartree equation in
We characterize the dynamics of the finite time blow up solutions with
minimal mass for the focusing mass critical Hartree equation with
data and data, where we make use of the
refined Gagliardo-Nirenberg inequality of convolution type and the profile
decomposition. Moreover, we also analyze the mass concentration phenomenon of
such blow up solutions.Comment: 25page
On local smoothing problems and Stein's maximal spherical means
It is proved that the local smoothing conjecture for wave equations implies
certain improvements on Stein's analytic family of maximal spherical means.
Some related problems are also discussed.Comment: 16pages, 1figur
Nonlinear Schr\"odinger equation with Coulomb potential
In this paper, we study the Cauchy problem for the nonlinear Schr\"odinger
equations with Coulomb potential with on . We
mainly consider the influence of the long range potential on the
existence theory and scattering theory for nonlinear Schr\"odinger equation. In
particular, we prove the global existence when the Coulomb potential is
attractive, i.e. and scattering theory when the Coulomb potential is
repulsive i.e. . The argument is based on the interaction Morawetz-type
inequalities and the equivalence of Sobolev norms.Comment: 28 page
Forward self-similar solutions of the fractional Navier-Stokes Equations
We study forward self-similar solutions to the 3-D Navier-Stokes equations
with the fractional diffusion First, we construct a
global-time forward self-similar solutions to the fractional Navier-Stokes
equations with for arbitrarily large self-similar initial
data by making use of the so called blow-up argument. Moreover, we prove that
this solution is smooth in . In particular, when
, we prove that the solution constructed by Korobkov-Tsai [Anal. PDE
9 (2016), 1811-1827] satisfies the decay estimate by establishing regularity of
solution for the corresponding elliptic system, which implies this solution has
the same properties as a solution which was constructed in [Jia and
\v{S}ver\'{a}k, Invent. Math. 196 (2014), 233-265].Comment: 46page
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