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Dlotko, Tomasz; Kania, Maria B.; Ma, Shan

doi:10.1016/j.jmaa.2013.10.007

Cited by 7 publications

(7 citation statements)

References 24 publications

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“…More related results can be found in previous studies [13][14][15][16][17][18][19][20][21][22][23][24][25][26] and the references therein. It is an important problem to prove that the global attractor has a finite fractal dimension (see Section 4 for a definition).…”

Section: Introductionmentioning

confidence: 62%

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The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

Wang

Huang

2020

Math Meth Appl Sci

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In this work, we are interested in the finite fractal dimensionality of the global attractor for a weakly damped Kordeweg‐de Vries (KdV) equation. For the KdV equation with periodic boundary condition, this is done by Ghidaglia in 1988. But since then, it seems little is known on this topic for the KdV equation on unbounded domains. The main difficulties are (a) the dissipative effect of the KdV equation is weak, and (b) the Sobolev embeddings on double-struckR are not compact. To overcome these difficulties, we present some new idea to prove the Chueshov‐Lasicka quasi‐stable estimates for the KdV equation on double-struckR. In this way, we show that for the KdV equation on the real line, the global attractor has a finite fractal dimension in the sharp space H3false(double-struckRfalse) whenever the force belongs to L2false(double-struckRfalse).

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Section: Introductionmentioning

confidence: 62%

“…From this, we use (18) and Proposition 3 to obtain that . This, together with (27), (34) implies that…”

Section: Lemma 2 Assume That H Satisfies Equation 30mentioning

confidence: 99%

The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

Wang

Huang

2020

Math Meth Appl Sci

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“…We need to use a more general than (4.11) expression (1.6) for the fractional powers, taken from [34, (3.4), p. 59], which states that: 12) where the original term (2) (α) (2−α) has been transformed using the known properties of the function;…”

Section: Moment Inequality Extendedmentioning

confidence: 99%

“…But such technique offers further possible generalizations, first to study the problems, like e.g. Korteweg-de Vries equation and its extensions [7,8,12,13], where the solutions are obtained as a limit of solutions to parabolic regularizations of such equations (the method known as vanishing viscosity technique, originated by Hopf, Oleinik, Lax in 1950th); see also [9,10]. Another possible application of Henry's technique is to study critical problems (e.g.…”

Section: Introductionmentioning

confidence: 99%

Navier–Stokes Equation and its Fractional Approximations

Dlotko

2016

Appl Math Optim

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“…Moreover, Dlotko and Sun showed that an ( H 1 ( R ), H 3 ( R )) global attractor exists. For dynamics of KdV‐B systems, we refer the readers to Dlotko et al Finally, Guo et al proved the existence of an ( H 2 ( R ), H 5 ( R )) global attractor of the fractional dissipative KdV equation.…”

Section: Introductionmentioning

confidence: 99%

Regular attractor for damped KdV‐Burgers equations on R

Guo

Wang

2017

Math Methods in App Sciences

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We study the well-posedness and dynamic behavior for the KdV-Burgers, as an application we obtain the local well-posedness. Then the global well-posedness follows from a uniform estimate for solutions as tgoes to infinity. Next, we prove the asymptotical regularity of solutions in space H 9 2 − 1 p − (R) and H 7 2 + 1 2q −,q (R) by the smoothing effect of e −t( 2 x − 3 x ) . The regularity and the asymptotical compactness in L 2 yields the asymptotical compactness in H 7 2 − 1 p ⋂ H 3,q (R)by an interpolation arguement. Finally, we conclude the existence of an (L 2 (R), H 7 2 − 1 p (R) ⋂ H 3,q (R)) globalattractor.

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Korteweg–de Vries–Burgers system inRN

Cited by 7 publications

References 24 publications

The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

The global attractor for the weakly damped KdV equation on R has a finite fractal dimension

Navier–Stokes Equation and its Fractional Approximations

Regular attractor for damped KdV‐Burgers equations on R

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