Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation

Dix, Daniel B.

doi:10.1016/0022-0396(91)90148-3

Cited by 45 publications

(22 citation statements)

References 7 publications

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“…When "1 we obtain the dispersion like that in the Benjamin}Ono}Burgers equation (cf. [17]). Our methods do not allow us to treat this case (here '1), nevertheless, we may observe change in the asymptotic expansion of solutions when approaches 1.…”

Section: Introductionmentioning

confidence: 99%

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Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

Karch

1999

Math. Meth. Appl. Sci.

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Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to the our classification. Furthermore, we given two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [DP18]. Contents 1. Introduction 1 2. The role of asymptotics 12 3. Finding critical exponents 20 4. Technical lemmas for decay 28 5. Testing the hunch: global existence 34 6. Blow-up results 46 References 54

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Section: Introductionmentioning

confidence: 99%

“…Concerning the decay of solutions to (1.1) or (1.2) for q"2 in various norms Dix [17] studied (1.1) for "1 and Amick et al [4]}(1.1) with "2 and (1.2). Next, Dix [18] developed the theory of equations more general than that in (1.1) for an integer q'2.…”

Section: Introductionmentioning

confidence: 99%

Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

Karch

1999

Math. Meth. Appl. Sci.

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“…Amick, Bona and Schonbek [4], Bona and Luo [5][6], Dix [8][9][10] and Zhang [15][16][17][18][19][20] established these results for various similar/simpler model equations. Theorem 1.2 is a summary of these known decay results.…”

Section: Theorem 11 (Existence and Uniqueness) Let The Initial Functionmentioning

confidence: 64%

Solutions to some open problems in Fluid dynamics

Zhang

2008

Chin. Ann. Math. Ser. B

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Let u = u(x, t, u0) represent the global solution of the initial value problem for the one-dimensional fluid dynamics equationwhere α > 0, β ≥ 0, γ ≥ 0, δ ≥ 0 and ε ≥ 0 are constants. This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations.The nonlinear function satisfies the conditions f (0) = 0, |f (u)| → ∞ as |u| → ∞, and f ∈ C 1 (R), and there exist the following limitsSuppose that the initial function u0 ∈ L 1 (R) ∩ H 2 (R). By using energy estimates, Fourier transform, Plancherel's identity, upper limit estimate, lower limit estimate and the results of the linear problemthe author justifies the following limits (with sharp rates of decay)where 0!! = 1, 1!! = 1 and m!! = 1 · 3 · · · · (2m − 3) · (2m − 1). Moreoverif the initial function u0(x) = ρ0 (x), for some function ρ0 ∈ C 1 (R) ∩ L 1 (R) and R ρ0(x)dx = 0.

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“…It is of concern to accent that Op p (a) generates a dense (DSG) of class A 2 as well as that −Op p (a) generates a dense (UDSG) of (p s )-class iff s ∈ (1, 2], −Op p (a) generates a dense (UDSG) of {p s }-class iff s ∈ (1, 2), and that −Op p (a) does not generate a (DSG). Finally, let us mention that L p -realizations of some special cases of pseudodifferential operators associated to the linearized Benjamin-Ono-Burgers equation [31,117] likewise generate ρ-hypoanalytic integrated (C-)semigroups. …”

Section: Theorem 37 Let a Be A Closed Linear Operator And Letmentioning

confidence: 99%

Differential and Analytical Properties of Semigroups of Operators

Kostić

2010

Integr. Equ. Oper. Theory

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We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23: [413][414][415][416][417][418][419][420][421][422][423][424][425][426][427][428][429] 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded 'commuting' perturbations.

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Temporal asymptotic behavior of solutions of the Benjamin-Ono-Burgers equation

Cited by 45 publications

References 7 publications

Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

Solutions to some open problems in Fluid dynamics

Differential and Analytical Properties of Semigroups of Operators

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