The Cauchy Problem for the Two Layer Viscous Shallow Water Equations
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
THE CAUCHY PROBLEM FOR THE TWO LAYER VISCOUS SHALLOW WATER EQUATIONS∗
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Pengcheng MU (
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China E-mail : [email protected]
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Qiangchang JU (
†
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China E-mail : ju [email protected] Abstract In this paper, the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data. The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs’ regularization method. Key words
two layer shallow water equations; global strong solution; hybrid Besov spaces
2010 MR Subject Classification
1
76N10; 35Q35; 35B30
Introduction
The two layer shallow water equations, which can be used to describe the interaction of Mediterranean and Atlantic water in the strait of Gibraltar [16], are written as follows [21]: h1t + div(h1 u1 ) = 0, ρ1 (h1 u1 )t + ρ1 div(h1 u1 ⊗ u1 ) + gρ1 h1 ∇h1 + gρ2 h1 ∇h2 −β h ∇(△h ) − β h ∇(△h ) = 2ν div(h · ∇u ), 1 2 1 2 1 1 1 1 1 (1.1) h2t + div(h2 u2 ) = 0, ρ2 (h2 u2 )t + ρ2 div(h2 u2 ⊗ u2 ) + gρ2 h2 ∇h1 + gρ2 h2 ∇h2 −β2 h2 ∇(△h1 ) − β2 h2 ∇(△h2 ) = 2ν2 div(h2 · ∇u2 ), (h , u , h , u )| 1 1 2 2 t=0 = (h10 , u10 , h20 , u20 ),
where index 1 refers to the deeper layer and index 2 the upper layer of the flow; ρ1 and ρ2 denote the densities and ρ2 < ρ1 ; ν1 and ν2 denote the viscosity coefficients; β1 and β2 denote the interface and free surface tension coefficients, respectively; and g is the gravitational ∗ Received
July 8, 2019; revised May 17, 2020. Ju was supported by the NSFC (11571046, 11671225), the ISF-NSFC joint research program NSFC (11761141008) and the BJNSF (1182004). † Corresponding author: Qiangchang JU.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
acceleration. All of these physical coefficients are positive constants. hj = hj (t, x) and uj = uj (t, x) denote the thickness and velocity field of each layer, where j = 1, 2. Distinguished from the single layer model, the two layer shallow water equations capture something of the density stratification of the ocean, and it is a powerful model of many geophysically interesting phenomena, as well as being physically realizable in the laboratory [4, 9, 19]. However, there are only a few mathematical analyse of the two layer model. Zabsonr´e-Reina [21] obtained the existence of global weak solutions in a periodic domain and Roamba-Zabsonr´e [18] proved the global existence of weak solutions for the two layer viscous shallow water equations without friction or capillary term. There are other results regarding weak solutions of the two layer shallow water equations in [10, 15]. To the best of our knowledge, there are no resu
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