Optimal Decay Rates to Conservation Laws With Diffusion-Type Terms of Regularity-Gain and Regularity-Loss

Duan, Renjun; Ruan, Lizhi; Zhu, Changjiang

doi:10.1142/s0218202512500121

Cited by 43 publications

(29 citation statements)

References 30 publications

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“…However, in order to obtain the long time behavior of the solutions, the authors assume that the initial data belong to some H N (R d ) spaces where N is large enough. In view of our analysis here we believe that the results in [7] can be obtained by assuming less regularity on the initial data. The analysis of these models by our methods remains to be considered in future papers.…”

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confidence: 89%

“…When q > 1 + 1/d or B = 0 1,d the asymptotic profile is the rescaled heat kernel solution of The models considered here could be related to the ones considered in [7], where a scalar conservation law with a diffusion-type source of the type u t + ∇·f (u) = ΔP s u is analyzed. There P s is essentially given by P s u(ξ) (1 + |ξ| 2 ) −sû (ξ) and even more general models are considered.…”

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confidence: 99%

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A Compactness Tool for the Analysis of Nonlocal Evolution Equations

Ignat¹,

Ignat²,

Stancu‐Dumitru³

2015

SIAM J. Math. Anal.

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In this paper we analyze the long time behavior of the solutions of a nonlocal diffusionconvection equation. We give a new compactness criterion in the Lebesgue spaces L p ((0, T ) × Ω) and use it to obtain the first term in the asymptotic expansion of the solutions. Previous results of [J. Bourgain, H. Brezis, and P. Mironescu, in Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001] are used to obtain a compactness result in the spirit of the AubinLions-Simon lemma. COMPACTNESS TOOLS FOR NONLOCAL EVOLUTION EQUATIONSSince J and G have mass one the mass conservation property holds:Since the proof of the global well-posedness follows by the same fixed-point argument as in [15] we will omit it here.Observe that under the assumptions (H0), (H1), and (H3) there exist positive constants R, δ such that|ξ| ≥ R, Downloaded 04/13/15 to 128.122.253.212. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php When q = 1 + 1/d and B = 0 1,d , U is the unique solution of the following equation:The well-posedness of this system has been analyzed in [8] in the one-dimensional case and in [9] in the multidimensional case. It has been proved in [1] that the profile f m is of constant sign and decays exponentially to zero as |x| → ∞.We remark that in the case of the symmetric function G, i.e., G(z) = G(−z), the solution of (1.1) converges to the heat kernel since in this case B vanishes. When B = 0 1,d we obtain in the limit the solutions of the viscous convection-diffusion equation. Throughout the paper we will consider the case of nonnegative initial data, so nonnegative solutions of system (1.1). The case of sign-change solutions could also be analyzed with small modifications of the proof (see [8] for a rigorous treatment of the critical case for the convection-diffusion equation).In the linear case, i.e., u t = J * u − u, the asymptotic behavior has been obtained in [5] by means of Fourier analysis techniques and in [19] by scaling methods. In [19] the scaling argument works since it is applied to the smooth part of the solution. Refined asymptotics have been obtained in [16,18]. We also recall here [31,32], where a scaling method is used for equations of the type u t = J * u − u − u p . There the authors obtain barriers for W and its derivatives, W being the smooth part of the solution of the linear equation u t = J * u − u. Once these barriers are obtained the authors split the solution of the nonlinear problem in a way that permits obtaining uniform Hölder estimates and then compactness. The method developed here is more flexible in the sense that it uses only energy estimates that involve the linear part of the equation and the good sign of the nonlinearity.In the local case, i.e., u t = Δu + a · ∇(|u| q−1 u), the same analysis has been performed in a series of papers. In [10] the case q ≥ 1 + 1/d is treated and the results in the critical case have been obtained by a careful space-time change of variables and using weighted Sobolev spaces. The subcritical...

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confidence: 89%

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confidence: 99%

A Compactness Tool for the Analysis of Nonlocal Evolution Equations

Ignat¹,

Ignat²,

Stancu‐Dumitru³

2015

SIAM J. Math. Anal.

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“…In this section, we will establish the global existence of solution to the compressible non‐isentropic Navier–Stokes–Maxwell system –. For later use and clear reference, the following sobolev inequality about the L p estimate on any two product terms with the sum of the order of their derivatives equal to a given integer is listed as follows .Lemma Let n ≥1. Let

α^{1} = ((), α_{1}^{1}, \dots, α_{n}^{1})

and

α^{2} = ((), α_{1}^{2}, \dots, α_{n}^{2})

be two multi‐indices with | α 1 |= k 1 ,| α 2 |= k 2 and set k = k 1 + k 2 .…”

Section: Asymptotic Behavior Of the Nonlinear Systemmentioning

confidence: 99%

Large time behavior for the non‐isentropic Navier–Stokes–Maxwell system

Liu

2016

Math Methods in App Sciences

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Communicated by C. MiaoIn this paper, we are concerned with the system of the non-isentropic compressible Navier-Stokes equations coupled with the Maxwell equations through the Lorentz force in three space dimensions. The global existence of solutions near constant steady states is established, and the time-decay rates of perturbed solutions are obtained. The proof for existence is due to the classical energy method, and the investigation of large-time behavior is based on the linearized analysis of the non-isentropic Navier-Stokes-Poisson equations and the electromagnetic part for the linearized isentropic NavierStokes-Maxwell equations. In the meantime, the time-decay rates obtained by Zhang, Li, and Zhu [J. Differential Equations, 250(2011), 866-891] for the linearized non-isentropic Navier-Stokes-Poisson equations are improved.

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“…(1.4) investigated in [3]. The authors studied the well-posedness and large time behavior for the Cauchy problem (1.4) for arbitrary space dimensions and for all s 1 ∈ R in the framework of small-amplitude classical solutions.…”

Section: Introductionmentioning

confidence: 99%