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1
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Derivation and well-posedness of Boussinesq/Boussinesq systems for internal waves

100%
Anh C.
Annales Polonici Mathematici
|
2009
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tom 96
|
nr 2
127-161
EN
We consider the propagation of internal waves at the interface between two layers of immiscrible fluids of different densities, under the rigid lid assumption, with the presence of surface tension and with uneven bottoms. We are interested in the case where the flow has a Boussinesq structure in both the upper and lower fluid domains. Following the global strategy introduced recently by Bona, Lannes and Saut [J. Math. Pures Appl. 89 (2008)], we derive an asymptotic model in this regime, namely the Boussinesq/Boussinesq systems. Then using a contraction-mapping argument and energy methods, we prove that the derived systems which are linearly well-posed are in fact locally nonlinearly well-posed in suitable Sobolev classes. We recover and extend some known results on asymptotic models and well-posedness, for both surface waves and internal waves.
2
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Finite-dimensional pullback attractors for parabolic equations with Hardy type potentials

64%
Anh C., Yen T.
Annales Polonici Mathematici
|
2011
|
tom 102
|
nr 2
161-186
EN
Using the asymptotic a priori estimate method, we prove the existence of a pullback 𝓓-attractor for a reaction-diffusion equation with an inverse-square potential in a bounded domain of $ℝ^{N}$ (N ≥ 3), with the nonlinearity of polynomial type and a suitable exponential growth of the external force. Then under some additional conditions, we show that the pullback 𝓓-attractor has a finite fractal dimension and is upper semicontinuous with respect to the parameter in the potential.
3
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Finite-dimensional Pullback Attractors for Non-autonomous Newton-Boussinesq Equations in Some Two-dimensional Unbounded Domains

64%
Anh C., Son D.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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tom 62
|
nr 3
265-289
EN
We study the existence and long-time behavior of weak solutions to Newton-Boussinesq equations in two-dimensional domains satisfying the Poincaré inequality. We prove the existence of a unique minimal finite-dimensional pullback $D_σ$-attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms.
4
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Global Attractors for a Class of Semilinear Degenerate Parabolic Equations on $ℝ^N$

64%
Anh C., Thuy L.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2013
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tom 61
|
nr 1
47-65
EN
We prove the existence of global attractors for the following semilinear degenerate parabolic equation on $ℝ^N$: ∂u/∂t - div(σ(x)∇ u) + λu + f(x,u) = g(x), under a new condition concerning the variable nonnegative diffusivity σ(·) and for an arbitrary polynomial growth order of the nonlinearity f. To overcome some difficulties caused by the lack of compactness of the embeddings, these results are proved by combining the tail estimates method and the asymptotic a priori estimate method.
5
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Pullback attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators

64%
Anh C., Bao T.
Annales Polonici Mathematici
|
2011
|
tom 101
|
nr 1
1-19
EN
Using the asymptotic a priori estimate method, we prove the existence of pullback attractors for nonautonomous quasilinear degenerate parabolic equations involving weighted p-Laplacian operators in bounded domains, without restriction on the growth order of the polynomial type nonlinearity and on the exponential growth of the external force. The results obtained improve some recent ones for nonautonomous reaction-diffusion equations. Moreover, a relationship between pullback attractors and uniform attractors is given.
6
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Long-time behavior for 2D non-autonomous g-Navier-Stokes equations

64%
Anh C., Quyet D.
Annales Polonici Mathematici
|
2012
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tom 103
|
nr 3
277-302
EN
We study the first initial boundary value problem for the 2D non-autonomous g-Navier-Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback $𝓓_σ$-attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force is time-independent and "small", the existence, uniqueness and global stability of a stationary solution are also studied.
7
Content available remote

Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators

64%
Anh C., Quang N.
Annales Polonici Mathematici
|
2010
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tom 98
|
nr 3
251-271
EN
We consider the first initial boundary value problem for nonautonomous quasilinear degenerate parabolic equations involving weighted p-Laplacian operators, in which the nonlinearity satisfies the polynomial condition of arbitrary order and the external force is normal. Using the asymptotic a priori estimate method, we prove the existence of uniform attractors for this problem. The results, in particular, improve some recent ones for nonautonomous p-Laplacian equations.
8
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Existence and upper semicontinuity of uniform attractors in $H¹(ℝ^N)$ for nonautonomous nonclassical diffusion equations

64%
Anh C., Toan N.
Annales Polonici Mathematici
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2014
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tom 111
|
nr 3
271-295
EN
We prove the existence of uniform attractors $𝓐_{ε}$ in the space $H¹(ℝ^N)$ for the nonautonomous nonclassical diffusion equation $u_t - ε Δu_t - Δu + f(x,u) + λu = g(x,t)$, ε ∈ [0,1]. The upper semicontinuity of the uniform attractors ${𝓐_{ε}}_{ε∈[0,1]}$ at ε = 0 is also studied.
9
Content available remote

Pullback attractors for non-autonomous 2D MHD equations on some unbounded domains

64%
Anh C., Son D.
Annales Polonici Mathematici
|
2015
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tom 113
|
nr 2
129-154
EN
We study the 2D magnetohydrodynamic (MHD) equations for a viscous incompressible resistive fluid, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the magnetic fields, in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality with a large class of non-autonomous external forces. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal pullback $D_σ$-attractor for the process associated to the problem. An upper bound on the fractal dimension of the pullback attractor is also given.
10
Content available remote

Inertial manifolds for retarded second order in time evolution equations in admissible spaces

64%
Anh C., Hieu L.
Annales Polonici Mathematici
|
2013
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tom 108
|
nr 1
21-42
EN
Using the Lyapunov-Perron method, we prove the existence of an inertial manifold for the process associated to a class of non-autonomous semilinear hyperbolic equations with finite delay, where the linear principal part is positive definite with a discrete spectrum having a sufficiently large distance between some two successive spectral points, and the Lipschitz coefficient of the nonlinear term may depend on time and belongs to some admissible function spaces.
11
Content available remote

Global existence and long-time behavior of solutions to a class of degenerate parabolic equations

64%
Anh C., Hung P.
Annales Polonici Mathematici
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2008
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tom 93
|
nr 3
217-230
EN
We study the global existence and long-time behavior of solutions for a class of semilinear degenerate parabolic equations in an arbitrary domain.
12
Content available remote

Global Attractor for a Class of Parabolic Equations with Infinite Delay

52%
Anh C., Hieu L., Thanh D.
Bulletin of the Polish Academy of Sciences. Mathematics
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2014
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tom 62
|
nr 1
49-60
EN
We prove the existence of a compact connected global attractor for a class of abstract semilinear parabolic equations with infinite delay.
13
Content available remote

On the global attractors for a class of semilinear degenerate parabolic equations

52%
Anh C., Binh N., Thuy L.
Annales Polonici Mathematici
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2010
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tom 98
|
nr 1
71-89
EN
We prove the existence and upper semicontinuity with respect to the nonlinearity and the diffusion coefficient of global attractors for a class of semilinear degenerate parabolic equations in an arbitrary domain.
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