Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations

Wu, Guochun; Zhang, Yinghui; Zou, Weiyuan

doi:10.1112/jlms.12393

Cited by 16 publications

(9 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The key idea here is that we will make full use of the benefit of low-frequency and high-frequency decomposition to close the second-order energy estimate. First, by applying Duhamel's principle, the key linear decay estimates in Wu-Zhang-Zou [26] and nonlinear energy estimates, we can get the optimal decay rate of ∇ 2 (m l , v l , B l ) L 2 in lemma 3.1. Second, we derive the optimal decay rate on ∇ 2 (m h , v h , B h ) L 2 .…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

Wang

Zhang

2022

Nonlinearity

Self Cite

View full text Add to dashboard Cite

We investigate time-decay rates of strong solutions to the 3D compressible magnetohydrodynamics equations with large initial data. The main novelty of this paper is two-fold: first, we prove the upper optimal decay rates of the higher-order spatial derivatives of the solution, which are the same as those of the heat equation, and faster than the decay rates in the previous related works. Second, if the initial data satisfy some additional low frequency assumption, we also show the lower optimal decay rates of the solution as well as its all-order spatial derivatives. Therefore, our decay rates are optimal in this sense. Our methods mainly involve the Fourier splitting method, low-frequency and high-frequency decomposition and delicate energy estimates.

show abstract

Section: Resultsmentioning

confidence: 99%

“…Under the assumption that the initial energy is suitably small, Li-Xu-Zhang [19], Liu-Yu-Zhang [20] and Suen-Hoff [23] proved the global existence of small energy weak or classic solutions respectively. Recently, Wu-Zhang-Zou [26] obtained the optimal decay rates of such small energy solutions.…”

Section: History Of the Problemmentioning

confidence: 99%

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

Wang

Zhang

2022

Nonlinearity

Self Cite

View full text Add to dashboard Cite

show abstract

“…Here we only list some of them, which are related to our study for the global well-posedness and decay estimates of the solutions of Cauchy problem. The interested readers can also refer to earlier studies 1,[17][18][19][20][21][22][23][24][25][26][27][28] for details.…”

Section: Introductionmentioning

confidence: 99%

The Cauchy problem for the nonisentropic compressible MHD fluids: Optimal time‐decay rates

Huang

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper is concerned with the time‐decay rates of the strong solutions of the three‐dimensional nonisentropic compressible magnetohydrodynamic (MHD) system. First, motivated by Pu and Guo's result [Z. Angew. Math. Phys. 64 (2013) 519–538], we establish the existence result of a unique local‐in‐time strong solution for the MHD system. Then, we derive a priori estimates and use the continuity argument to obtain the global‐in‐time solution, where the initial perturbation is small in H2$$ {H}^2 $$‐norm. Finally, based on Fourier theory and the idea of cancelation of a low‐medium frequent part as in [Sci. China Math. 65 (2022) 1199–1228], we get the optimal time‐decay rates (including highest‐order derivatives) of strong solutions for nonisentropic MHD fluids when the boundedness of L1$$ {L}^1 $$‐norm of the initial perturbation is required. Our result is the first one concerning with the optimal decay estimates of the highest‐order derivatives of the nonisentropic MHD system.

show abstract

“…Moreover, the uniqueness and continuous dependence of these weak solutions were established in [43] by modifying the initial conditions. Meanwhile, Wu-Zhang-Zou [46] obtained the optimal time-decay rates of weak solutions with discontinuous initial data, i.e., for t ≥ 1,…”

Section: Introductionmentioning

confidence: 99%

“…if (ρ 0 − ρ, u 0 , b 0 ) L 1 is bounded. Apart from large-energy weak solutions [15] and intermediate weak solutions [36,43,44,46], the third type of solutions to (1.5) are the small-smooth solutions. More precisely, Li-Xu-Zhang [26] showed the global existence of classical solutions to the 3D Cauchy problem of (1.5) with small initial energy but possibly large oscillations and vacuum states at both the interior domain and the far field which generalized the previous result proved by Huang-Li-Xin [17] for compressible Navier-Stokes system.…”

Section: Introductionmentioning

confidence: 99%

Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity

Liu¹,

Zhong²

2022

Preprint

View full text Add to dashboard Cite

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In particular, it is unknown whether the entropy remains its boundedness. In the present paper, we investigate the Cauchy problem to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity only. We show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. The main tools are based on singularly weighted energy estimates and De Giorgi type iteration techniques developed by Li and Xin (https://arxiv.org/abs/2111.14057) for the 3D full compressible Navier-Stokes system. Some new mathematical techniques and useful estimates are developed to deduce the lower and upper bounds on the entropy.

show abstract

Optimal time‐decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations

Cited by 16 publications

References 36 publications

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

Optimal large time behaviour of the 3D compressible magnetohydrodynamics equations with large initial data

The Cauchy problem for the nonisentropic compressible MHD fluids: Optimal time‐decay rates

Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity

Contact Info

Product

Resources

About