Large time behavior of solutions to the isentropic compressible fluid models of Korteweg type in R³

In this study, we discuss some limit analysis of a viscous capillary model of plasma, which is expressed as a so‐called the compressible Navier‐Stokes‐Poisson‐Korteweg equation. First, the existence of global smooth solutions for the initial value problem to the compressible Navier‐Stokes‐Poisson‐Korteweg equation with a given Debye length λ and a given capillary coefficient κ is obtained. We also show the uniform estimates of global smooth solutions with respect to the Debye length λ and the capillary coefficient κ. Then, from Aubin lemma, we show that the unique smooth solution of the 3‐dimensional Navier‐Stokes‐Poisson‐Korteweg equations converges globally in time to the strong solution of the corresponding limit equations, as λ tends to zero, κ tends to zero, and λ and κ simultaneously tend to zero. Moreover, we also give the convergence rates of these limits for any given positive time one by one.

“…Proof of Theorem Let ( ρ λ , u λ , ϕ λ )( x , t ) and

false(ρ_{1}^{0}, u_{1}^{0}, ϕ_{1}^{0} false) false(x, t false)

be the solution of and , respectively. It follows from Theorem that

\begin{matrix} alignleft \\ align-1 & align-2 ‖ (ρ^{normalλ} - 1, u^{normalλ}, ϕ^{normalλ}) (\cdot, t) {false‖}_{3} + \sum_{| α | = 4} [κ ‖ \partial_{x}^{α} (ρ^{normalλ} - 1) (\cdot, t) ‖ + {normalλ}^{2} ‖ \partial_{x}^{α} ϕ (\cdot, t) {false‖}_{1}^{2}] \\ align-1 & align-2 \leq C (‖ ρ_{0} - 1 {false‖}_{4} + ‖ u_{0} {false‖}_{3}) . \end{matrix}

Moreover, noting the initial data of are same as those of , and ‖ ρ 0 − 1‖ 4 + ‖ u 0 ‖ 3 is small enough, from the results of Tan and Wang, we also have

false‖ false(ρ_{1}^{0} - 1, u_{1}^{0} false(\cdot, t false) ‖_{3} + \sum_{false| α false| = 4} κ false‖ \partial_{x}^{α} false(ρ_{1}^{0} - 1 false) false(\cdot, t false) false‖ \leq C false(false‖ ρ_{0} - 1 ‖_{4} + false‖ u_{0} ‖_{3} false) .

From Theorem , we have

ϕ

…”

Section: Proof Of Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Moreover, noting the initial data of (4) are same as those of (2), and ‖ρ 0 − 1‖ 4 + ‖u 0 ‖ 3 is small enough, from the results of Tan and Wang, 21 we also have…”

mentioning

confidence: 96%

See 2 more Smart Citations

Some limit analysis of a three dimensional viscous compressible capillary model for plasma

Peng

Wang

2018

“…The problem (1) has attracted lots of mathematicians and physician's interests, and some important results were made, we may refer to other works. [4][5][6][7][8][9][10][11][12][13][14][15][16][17] Local existence and global existence of classical solutions in Sobolev space were established by Hattori and Li. 9 Global well posedness in the critical Besov spaces for initial data close enough to stable equilibria has been established by Danchin and Desjardins.…”

Section: Introductionmentioning

confidence: 99%

Optimal decay estimate of mild solutions to the compressible Navier‐Stokes‐Korteweg system in the critical Besov space

Wang

2018

In this paper, we consider the initial value problem for the compressible Navier-Stokes-Korteweg system in several space variables. Optimal decay estimate of mild solutions in the critical Besov spaces is established. The proof is based on the decay estimate of solutions operator in the low-frequency region and the energy estimate in the high-frequency region. KEYWORDScompressible Navier-Stokes-Korteweg equations, decay estimate, fourth-order wave equation with strong damping, mild solutions 9592

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