1,766 research outputs found
Well-posedness and global existence of 2D viscous shallow water system in Besov spaces
In this paper we consider the Cauchy problem for 2D viscous shallow water
system in Besov spaces. We firstly prove the local well-posedness of this
problem in , , by using the Littlewood-Paley theory, the Bony decomposition and the
theories of transport equations and transport diffusion equations. Then we can
prove the global existence of the system with small enough initial data in
, , and
. Our obtained results generalize and cover the recent results
in \cite{W}
Global weak solutions to a weakly dissipative HS equation
This paper is concerned with global existence of weak solutions for a weakly
dissipative HS equation by using smooth approximate to initial data and
Hellys theorem
Global Well-posedness for the Generalized Navier-Stokes System
In this paper we investigate well-posedness of the Cauchy problem of the
three dimensional generalized Navier-Stokes system. We first establish local
well-posedness of the GNS system for any initial data in the Fourier-Herz space
. Then we show that if the norm of the initial data is
smaller than C in the GNS system where is the viscosity coefficient,
the corresponding solution exists globally in time. Moreover, we prove global
well-posedness of the Navier-Stokes system without norm restrictions on the
corresponding solutions provided the norm of the initial data is
less than Our obtained results cover and improve recent results in
\cite{Zhen Lei,wu}
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations
This paper is concerned with a class of nonlocal dispersive models -- the
-equation proposed by H. Liu [ On discreteness of the Hopf equation,
{\it Acta Math. Appl. Sin.} Engl. Ser. {\bf 24}(3)(2008)423--440]: including integrable equations such as the
Camassa-Holm equation, , and the Degasperis-Procesi equation,
, as special models. We investigate both global regularity of
solutions and wave breaking phenomena for . It is shown
that as increases regularity of solutions improves: (i) , the solution will blow up when the momentum of initial data satisfies
certain sign conditions; (ii) , the solution will blow
up when the slope of initial data is negative at one point; (iii) and , global existence
of strong solutions is ensured. Moreover, if the momentum of initial data has a
definite sign, then for any global smoothness of the
corresponding solution is proved. Proofs are either based on the use of some
global invariants or based on exploration of favorable sign conditions of
quantities involving solution derivatives. Existence and uniqueness results of
global weak solutions for any are also presented. For
some restricted range of parameters results here are equivalent to those known
for the equations [e.g. J. Escher and Z. Yin, Well-posedness, blow-up
phenomena, and global solutions for the b-equation, {\it J. reine angew.
Math.}, {\bf 624} (2008)51--80.]Comment: 21 page
Local well-posedness and blow-up criteria for a two-component Novikov system in the critical Besov space
In this paper we mainly investigate the Cauchy problem of a two-component
Novikov system. We first prove the local well-posedness of the system in Besov
spaces with
by using the
Littlewood-Paley theory and transport equations theory. Then, by virtue of
logarithmic interpolation inequalities and the Osgood lemma, we establish the
local well-posedness of the system in the critical Besov space
. Moreover, we present two
blow-up criteria for the system by making use of the conservation laws.Comment: arXiv admin note: text overlap with arXiv:1505.0008
Global Weak Solution for a generalized Camassa-Holm equation
In this paper we mainly investigate the Cauchy problem of a generalized
Camassa-Holm equation. First by this relationship between the
Degasperis-Procesi equation and the generalized Camassa-Holm equation, we then
obtain two global existences result and two blow-up result. Then, we prove the
existence and uniqueness of global weak solutions.Comment: arXiv admin note: text overlap with arXiv:1505.00086,
arXiv:1511.0231
Analyticity of the Cauchy problem and persistence properties for a generalized Camassa-Holm equation
This paper is mainly concerned with the Cauchy problem for a generalized
Camassa-Holm equation with analytic initial data. The analyticity of its
solutions is proved in both variables, globally in space and locally in time.
Then, we present a persistence property for strong solutions to the system.
Finally, explicit asymptotic profiles illustrate the optimality of these
results.Comment: arXiv admin note: substantial text overlap with arXiv:1505.00086;
text overlap with arXiv:1202.0718 by other author
On the Cauchy problem of a weakly dissipative HS equation
In this paper, we study the Cauchy problem of a weakly dissipative HS
equation. We first establish the local well-posedness for the weakly
dissipative HS equation by Kato's semigroup theory. Then, we derive the
precise blow-up scenario for strong solutions to the equation. Moreover, we
present some blow-up results for strong solutions to the equation. Finally, we
give two global existence results to the equation
Global well-posedness for Euler-Nernst-Planck-Possion system in dimension two
In this paper, we study the Cauchy problem of the Euler-Nernst-Planck-Possion
system. We obtain global well-posedness for the system in dimension for
any initial data in under certain conditions of and .Comment: arXiv admin note: substantial text overlap with arXiv:1406.369
Global solution to liquid crystal flows in three dimensions
In this paper, we mainly study a hydrodynamic system modeling the flow of
nematic liquid crystals. In three dimensions, we first establish local
well-posedness of the initial-boundary value problem of the system. Then, we
prove the existence of global strong solution to the system with small
initial-boundary condition
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