| تعداد نشریات | 32 |
| تعداد شمارهها | 861 |
| تعداد مقالات | 8,364 |
| تعداد مشاهده مقاله | 53,003,990 |
| تعداد دریافت فایل اصل مقاله | 9,367,004 |
On the regularity theory for quasilinear elliptic systems with the application of Leray-Schauder method | ||
| Journal of Mathematical Modeling | ||
| مقاله 23، دوره 13، شماره 1، خرداد 2025، صفحه 169-182 اصل مقاله (187.71 K) | ||
| نوع مقاله: Research Article | ||
| شناسه دیجیتال (DOI): 10.22124/jmm.2024.27119.2387 | ||
| نویسنده | ||
| Mykola Yaremenko* | ||
| The National Technical University of Ukraine, Igor Sikorsky Kyiv Polytechnic Institute, 37, Prospect Beresteiskyi (former Peremohy), Kyiv, Ukraine, 03056 | ||
| چکیده | ||
| In this article, we consider an elliptic system of partial differential equations in the general form \[\sum _{i=1,..., n}\frac{d}{dx_{i} } A_{i} \left(x,\; \overrightarrow{u},\; \nabla \overrightarrow{u}\right) +B\left(x,\; \overrightarrow{u},\; \nabla \overrightarrow{u}\right)=0\] under fair general conditions on its structural coefficients. We study the regularity properties of the solutions to this system, and we establish the existence of a Holder solution by the modified Leray-Schauder fixed-point method and the application of the apriori estimations obtained with utilization of form-boundary conditions. | ||
| کلیدواژهها | ||
| Quasilinear Partial Differential Equation؛ Holder solution؛ regularity theory؛ Leray-Schauder theorem؛ form-bounded | ||
| مراجع | ||
|
[1] M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient L2-norm, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 757–762. [2] H. Amann, Invariant sets and existence theorems for semilinear parabolic and elliptic systems, J. Math. Anal. Appl. 65 (1978) 432–467. [3] C. Bao, B. Cui, X. Lou, W. Wu, and B. Zhuang, Fixed-time stabilization of boundary controlled lin- ear parabolic distributed parameter systems with space-dependent reactivity, IET Control Theory Appl. 15 (2020) 652–667. [4] C. Chen, R.M. Strain, H. Yau, T. Tsai, Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II, Commun. Partial Differ. Equ. 34 (2009) 203–232. [5] E. DeGiorgi, Sulla differenziabilita e lanaliticita delle estremali degli integrali multipli regolari, Mem. Accad Sc. Torino, C. Sc. Fis. Mat. Natur. 3 (1957) 25-43. [6] E. DeGiorgi, Un esempio di estremali discontinue per un problema variazionale di tipo ellitico, Bull. UMI 4 (1968) 135-137. [7] H. Dong, S. Kim, S. Lee, Estimates for fundamental solutions of parabolic equations in non- divergence form, J. Differ. Equ. 340 (2022) 557–591. [8] J.A. Goldstein, Q.S. Zhang, Linear parabolic equations with strongly singular potentials, Trans. Amer. Math. Soc. 355 (2002) 197–211. [9] D. Hongjie, L. Escauriaza, S. Kim, On C1/2,1, C1,2, and C0,0 estimates for linear parabolic operators, J. Evol. Equ. 21 (2021) 4641–4702. [10] S. Kim, S. Lee, Estimates for Green’s functions of elliptic equations in non-divergence form with continuous coefficients, Ann. Appl. Math. 37 (2021) 111–130. [11] D. Kinzebulatov, Y. A. Semenov, Heat kernel bounds for parabolic equations with singular (form- bounded) vector fields, Preprint, 2021, arXiv:2103.11482. [12] M. Kassmann, M. Weidner, The parabolic Harnack inequality for nonlocal equations, 2023, arXiv:2303.05975. [13] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Amer. Math. Soc. 23 (1968). [14] M. Li, W. Mao, Finite-time bounded control for coupled parabolic PDE-ODE systems subject to boundary disturbances, Math. Probl. Eng. 2020 (2020) Article ID 8882382. [15] Z. Qian, G. Xi, Parabolic equations with singular divergence-free drift vector fields, J. London Math. Soc. 100 (2019) 17–40. [16] Y. Tao, M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis- haptotaxis model with the remodeling of non-diffusible attractant, J. Differ. Equ. 257 (2014) 784– 815. [17] Y. Tao, Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis, Nonl. Aanl. RWA 11 (2010) 2056–2064. [18] Y. Tao, M. Winkler, Energy-type estimates and global solvability in a two-dimensional chemotaxis- haptotaxis model with the remodeling of non-diffusible attractant, J. Differ. Equ. 257 (2014) 784– 815. [19] A. Veretennikov, On weak existence of solutions of degenerate McKean-Vlasov equations, Preprint, 2023, ArXiv 2301.01532. [20] F.B. Weissler, Single-point blow-up for a semilinear initial value problem, J. Diff. Equ. 5 (1984) 204–224. [21] S. Wang, Y. Teng, P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics-informed neural networks, SIAM J. Sci. Comput. 43 (2021) A3055–A3081. [22] W.L.J. Wang, W. Guo, A backstepping approach to adaptive error feedback regulator design for one-dimensional linear parabolic PIDEs, J. Math. Anal. Appl. 503 (2021) Article ID 125310. [23] M.I. Yaremenko, The existence of a solution of evolution and elliptic equations with singular coefficients, Asian Res. J. Math. 15 (2017) 172–204. [24] M.I. Yaremenko, Quasi-linear evolution, and elliptic equations, J. Progressive Res. Math. 11 (2017) 1645–1669. [25] M.I. Yaremenko, The sequence of semigroups of nonlinear operators and their applications to study the Cauchy problem for parabolic equations, Sci. J. Ternopil Nat. Tech. Univ. 4 (2016) 149–160. [26] D. Zhang, L. Guo, G. E. Karniadakis, Learning in modal space: Solving time-dependent stochastic PDEs using physics-informed neural networks, SIAM J. Sci. Comput. 42 (2020) A639–A665. [27] Z.-Q. Chen, T. Kumagai, J. Wang, Stability of heat kernel estimates for symmetric non-local Dirich- let forms, Mem. Amer. Math. Soc. 271 2021 1–100. [28] S. Zhang, Z.-Q. Chen, Stochastic maximum principle for fully coupled forward-backward stochastic differential equations driven by sub-diffusion, Preprint, 2023. | ||
|
آمار تعداد مشاهده مقاله: 234 تعداد دریافت فایل اصل مقاله: 345 |
||