一类耗散型电流体动力学方程组自相似解的渐近稳定性
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摘要
主要考虑一类来源于电流体动力学中的由非线性非局部方程组耦合而成的耗散型系统的初值问题.利用Lorentz空间中广义L~p-L~q热半群估计和广义Hardy-Littlewood-Sobolev不等式,首先证明了该系统在Lorentz空间中自相似解的整体存在性和唯一性,然后建立了自相似解当时间趋于无穷时的渐近稳定性.因为Lorentz空间包含了具有奇性的齐次函数,因次上述结果保证了具有奇性的初值所对应的自相似解的整体存在性和渐近稳定性.
The authors consider a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics in the whole space R~n,n≥3. By making use of the generalized L~p-L~q heat semigroup estimates in the Lorentz spaces and the generalized Hardy-Littlewood-Sobolev inequality, the authors first prove global existence and uniqueness of self-similar solution in the Lorentz spaces, then establish the asymptotic stability of selfsimilar solutions as time goes to infinity. Since the authors Cope with the initial data in the Lorentz spaces, the existence of self-similar solutions provided the initial data are small homogeneous functions.
The authors consider a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics in the whole space R~n,n≥3. By making use of the generalized L~p-L~q heat semigroup estimates in the Lorentz spaces and the generalized Hardy-Littlewood-Sobolev inequality, the authors first prove global existence and uniqueness of self-similar solution in the Lorentz spaces, then establish the asymptotic stability of selfsimilar solutions as time goes to infinity. Since the authors Cope with the initial data in the Lorentz spaces, the existence of self-similar solutions provided the initial data are small homogeneous functions.
引文
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[2] Debye P, Hiickel E. Zur theorie der elektrolyte, II:das grenzgesetz fiir die elektrische leitfahigkeit[J]. Phys Z, 1923, 24:305-325.
[3] Mock M S. An initial value problem from semiconductor device theory[J]. SIAM J Math Anal, 1974, 5:597-612.
[4] Mock M S. Asymptotic behavior of solutions of transport equations for semiconductor devices[J]. J Math Anal Appl, 1975, 49:215-225.
[5] Biler P, Hebisch W, Nadzieja T. The Debye system:existence and large time behavior of solutions[J]. Nonlinear Anal, 1994, 23:1189-1209.
[6] Biler P, Dolbeault J. Long time behavior of solutions to Nernst-Planck and DebyeHiickel drift-diffusion systems[J]. Ann Henri Poincare, 2000, 1:461-472.
[7] Karch G. Scaling in nonlinear parabolic equations[J]. J Math Anal Appl, 1999,234:534-558.
[8] Kurokiba M, Ogawa T. Well-posedness for the drift-diffusion system in L~p arising from the semiconductor device simulation[J]. J Math Anal Appl, 2008, 342:1052-1067.
[9] Ogawa T, Shimizu S. The drift-diffusion system in two-dimensional critical Hardy space[J]. J Funct Anal, 2008, 255:1107-1138.
[10] Ogawa T, Shimizu S. End-point maximal regularity and wellposedness of the two dimensional Keller-Segel system in a critical Besov space[J]. Math Z, 2010, 264:601-628.
[11] Zhao J H, Liu Q, Cui S B. Existence of solutions for the Debye-Huckel system with low regularity initial data[J]. Acta Applicanda Mathematicae, 2013, 125:1-10.
[12] Jerome J W. Analytical approaches to charge transport in a moving medium[J]. Tran Theo Stat Phys, 2002, 31:333-366.
[13] Jerome J W, Sacco R. Global weak solutions for an incompressible charged fluid with multi-scale couplings:initial-boundary-value problem[J]. Nonlinear Anal, 2009,71:2487-2497.
[14] Ryham R J. Existence, uniqueness, regularity and long-term behavior for dissipative systems modeling electrohydrodynamics[J]. Analysis of PDEs, 2009:1-60, arXiv:0910.4973v1.
[15] Schmuck M. Analysis of the Navier-Stokes-Nernst-Planck-Poisson system[J]. Math Models&Methods Appl Sci, 2009, 19(6):993-1015.
[16] Zhao J H, Deng C, Cui S B. Well-posedness of a dissipative system modeling electrohydrodynamics in Lebesgue spaces[J]. Diff Equa Appl, 2011, 3(3):427-448.
[17] Zhang Z,Yin Z Y. Global well-posedness for the Navier-Stokes-Nernst-Planck-Poisson system in dimension two[J]. Applied Mathematics Letters, 2015, 40:102-106.
[18] Zhao J H, Liu Q. Well-posedness and decay for the dissipative system modeling electrohydrodynamics in negative Besov spaces[J]. J Differential Equations, 2017, 263:1293-1322.
[19] Fan J S, Gao H J. Uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics[J]. Nonlinear Anal, 2009, 70:2382-2386.
[20] Fan J S, Li F C, Nakamura G. Regularity criteria for a mathematical model for the deformation of electrolyte droplets[J]. Applied Mathematics Letters, 2013, 26:494-499.
[21] Fan J S, Nakamura G, Zhou Y. On the Cauchy problem for a model of electro-kinetic fluid[J]. Applied Mathematics Letters, 2012, 25:33-37.
[22] Zhao J H, Bai M. Blow-up criteria for the three dimensional nonlinear dissipative system modeling electro-hydrodynamics[J]. Nonlinear Analysis:Real World Applications,2016, 31:210-226.
[23] Tian G, Xin Z P. One-point singular solutions to the Navier-Stokes equations[J]. Topol Methods Nonl Anal, 1998, 11:135-145.
[24] Barraza O A. Self-similar solutions in weak L~p spaces of the Navier-Stokes equations[J]. Rev Mat Iberoamericana, 1996, 12:411-439.
[25] Barraza O A. Regularity and stability for the solutions of the Navier-Stokes equations in Lorentz spaces[J]. Nonlinear Anal, 1999, 35:747-764.
[26] Yamazaki M. The Navier-Stokes equations in the weak-L~n spaces with time-dependent external force[J]. Math Ann, 2000, 317:635-675.
[27] Stein E, Weiss G. Introduction to Fourier analysis on euclidean spaces[M]. Princeton:Princeton University Press, 1971.
[28] Meyer Y. Wavelets, paraproducts and Navier-Stokes equations[M]//Current developments in Mathematics, Cambridge:International Press, 1996:105-212.
[29] Bergh J, Lofstrom J. Interpolation spaces[M]. An Introduction, Berlin:SpringerVerlag, 1976.
[30] O'Neil R. Convolution operators and L~((p,q))spaces[J]. Duke Math J, 1963, 30:129-142.