# Article

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Keywords:
nonlinear parabolic systems; regularity problem
Summary:
Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.
References:
[1] Arkhipova A. A.: On the regularity of the solutions of the Neumann problem for quasilinear parabolic systems. Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), no. 5, 3–25 (Russian); translation in Russian Acad. Sci. Izv. Math. 45 (1995), no. 2, 231–253. MR 1307308
[2] Arkhipova A. A.: Reverse Hölder inequalities with boundary integrals and $L_p$-estimates for solutions of nonlinear elliptic and parabolic boundary value problems. Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2, 164, Adv. Math. Sci., 22, Amer. Math. Soc. (1995), 15–42. MR 1334137
[3] Arkhipova A. A.: Regularity of weak solutions to the model Venttsel problem for linear parabolic systems with nonsmooth in time principal matrix: $A(t)$-caloric approximation method. Manuscripta Math. 151 (2016), no. 3–4, 519–548. DOI 10.1007/s00229-016-0838-y | MR 3556832
[4] Arkhipova A. A.: Regularity of solutions of the model Venttsel' problem for quasilinear parabolic systems with nonsmooth in time principal matrices. Comput. Math. Math. Phys. 57 (2017), no. 3, 476–496. DOI 10.1134/S0965542517030034 | MR 3636034
[5] Arkhipova A. A., Stará J.: Boundary partial regularity for solutions of quasilinear parabolic systems with non smooth in time principal matrix. Nonlinear Anal. 120 (2015), 236–261. MR 3348057
[6] Arkhipova A. A., Stará J.: Regularity of weak solutions to linear and quasilinear parabolic systems of non-divergence type with non-smooth in time principal matrix: $A(t)$-caloric method. Forum Math. 29 (2017), no. 5, 1039–1064. DOI 10.1515/forum-2015-0222 | MR 3692026
[7] Arkhipova A. A., Stará J.: Regularity problem for $2m$-order quasilinear parabolic systems with non smooth in time principal matrix. $(A(t),m)$-caloric approximation method. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 111–146. MR 3867982
[8] Arkhipova A. A., Stará J., John O.: Partial regularity for solutions of quasilinear parabolic systems with nonsmooth in time principal matrix. Nonlinear Anal. 95 (2014), 421–435. MR 3130534
[9] Bögelein V., Duzaar F., Mingione G.: The boundary regularity of non-linear parabolic systems. I. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 1, 201–255. DOI 10.1016/j.anihpc.2009.09.003 | MR 2580509
[10] Campanato S.: Equazioni paraboliche del secondo ordine e spazi $L^{2,\theta}(\Omega,\delta)$. Ann. Mat. Pura Appl. (4) 73 (1966), 55–102 (Italian). DOI 10.1007/BF02415082 | MR 0213737
[11] Campanato S.: On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions. Ann. Mat. Pura Appl. (4) 137 (1984), 83–122. MR 0772253
[12] Diening L., Lengeler D., Stroffolini B., Verde A.: Partial regularity for minimizers of quasi-convex functionals with general growth. SIAM J. Math. Anal. 44 (2012), no. 5, 3594–3616. DOI 10.1137/120870554 | MR 3023424
[13] Diening L., Schwarzacher S., Stroffolini B., Verde A.: Parabolic Lipschitz truncation and caloric approximation. Calc. Var. Partial Differential Equations 56 (2017), no. 4, Art. 120, 27 pages. DOI 10.1007/s00526-017-1209-6 | MR 3672391
[14] Dong H., Kim D.: Parabolic and elliptic systems with VMO coefficients. Methods Appl. Anal. 16 (2009), no. 3, 365–388. MR 2650802
[15] Dong H., Kim D.: Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth. Communic. Partial Differential Equations 36 (2011), no. 10, 1750–1777. DOI 10.1080/03605302.2011.571746 | MR 2832162
[16] Dong H., Kim D.: On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients. Arch. Ration. Mech. Anal. 199 (2011), no. 3, 889–941. DOI 10.1007/s00205-010-0345-3 | MR 2771670
[17] Duzaar F., Grotowski J. F.: Optimal interior partial regularity for nonlinear elliptic systems: the method of $A$-harmonic approximation. Manuscripta Math. 103 (2000), no. 3, 267–298. DOI 10.1007/s002290070007 | MR 1802484
[18] Duzaar F., Mingione G.: Second order parabolic systems, optimal regularity, and singular sets of solutions. Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 6, 705–751. DOI 10.1016/j.anihpc.2004.10.011 | MR 2172857
[19] Duzaar F., Steffen K.: Optimal interior and boundary regularity for almost minimizers to elliptic variational integrals. J. Reine Angew. Math. 546 (2002), 73–138. MR 1900994
[20] Gehring F. W.: The $L^p$-integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973), 265–277. DOI 10.1007/BF02392268 | MR 0402038
[21] Giaquinta M., Giusti E.: Partial regularity for solutions to nonlinear parabolic systems. Ann. Mat. Pura Appl. (4) 97 (1973), 253–266. DOI 10.1007/BF02414914 | MR 0338568
[22] Giaquinta M., Struwe M.: On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179 (1982), no. 4, 437–451. DOI 10.1007/BF01215058 | MR 0652852
[23] Kinnunen J., Lewis J. L.: Higher integrability for parabolic systems of p-Laplacian type. Duke Math. J. 102 (2000), no. 2, 253–271. DOI 10.1215/S0012-7094-00-10223-2 | MR 1749438
[24] Krylov N. V.: Parabolic and elliptic equations with VMO coefficients. Comm. Partial Differential Equations 32 (2007), no. 1–3, 453–475. DOI 10.1080/03605300600781626 | MR 2304157
[25] Krylov N. V.: Lectures on Elliptic and Parabolic Equations in Sobolev Spaces. Graduate Studies in Mathematics, 96, American Mathematical Society, Providence, 2008. DOI 10.1090/gsm/096 | MR 2435520
[26] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N.: Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, 1968. DOI 10.1090/mmono/023 | MR 0241822
[27] Mingione G.: The singular set of solutions to non differentiable elliptic systems. Arch. Ration. Mech. Anal. 166 (2003), no. 4, 287–301. DOI 10.1007/s00205-002-0231-8 | MR 1961442

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