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Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts

  • Tuoc Phan (a1)
Abstract

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ( $x,t$ )-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

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This research is supported by the Simons Foundation, grant #354889

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[1] Acerbi, E. and Mingione, G., Gradient estimates for a class of parabolic systems . Duke Math. J. 136(2007), no. 2, 285320. https://doi.org/10.1215/S0012-7094-07-13623-8.
[2] Aronson, D. G. and Serrin, J., Local behavior of solutions of quasilinear parabolic equations . Arch. Rational Mech. Anal. 25(1967), 81122. https://doi.org/10.1007/BF00281291.
[3] Baroni, P., Lorentz estimates for degenerate and singular evolutionary systems . J. Differential Equations 255(2013), no. 9, 29272951. https://doi.org/10.1016/j.jde.2013.07.024.
[4] Berestycki, H., Hamel, F., and Nadirashvili, N., Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena . Commun. Math. Phys. 253(2005), 451480. https://doi.org/10.1007/s00220-004-1201-9.
[5] Bölegein, V., Global Calderón–Zygmund theory for nonlinear parabolic system . Calc. Var. Partial Differential Equations 51(2014), 555596. https://doi.org/10.1007/s00526-013-0687-4.
[6] Bui, T. A. and Duong, X. T., Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains . Calc. Var. Partial Differential Equations 56(2017), no. 2, Art. 47.
[7] Byun, S.-S., Palagachev, D. K., and Shin, P., Global Sobolev regularity for general elliptic equations of p-Laplacian type. arxiv:1703.09918.
[8] Byun, S.-S. and Wang, L., Parabolic equations with BMO nonlinearity in Reifenberg domains . J. Reine Angew. Math. 615(2008), 124. https://doi.org/10.1515/CRELLE.2008.007.
[9] Byun, S.-S. and Wang, L., W 1, p -regularity for the conormal derivative problem with parabolic BMO nonlinearity in Reifenberg domains . Discrete Contin. Dyn. Syst. 20(2008), no. 3, 617637.
[10] Caffarelli, L. A. and Peral, I., On W 1, p estimates for elliptic equations in divergence form . Comm. Pure Appl. Math. 51(1998), no. 1, 121. http://dx.doi.org/10.1002/(SICI)1097-0312(199801)51:1 <1::AID-CPA1 >3.0.CO;2-G.
[11] Caffarelli, L. A. and Vasseur, A., Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation . Ann. of Math. (2) 171(2010), no. 3, 19031930. https://doi.org/10.4007/annals.2010.171.1903.
[12] Chiarenza, F., Frasca, M., and Longo, P., Interior W 2, p estimates for nondivergence elliptic equations with discontinuous coefficients . Ricerche Mat. 40(1991), 149168.
[13] Dibenedetto, E., Degenerate parabolic equations. Universitext, Springer-Verlag, New York, 1993.
[14] Di Fazio, G., L p estimates for divergence form elliptic equations with discontinuous coefficients . Boll. Un. Mat. Ital. A 10(1996), no. 2, 409420.
[15] Dolzmann, G., Hungerbühler, N., and Müller, S., Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side . J. Reine Angew. Math. 520(2000), 135. https://doi.org/10.1515/crll.2000.022.
[16] Dong, H. and Kim, D., Global regularity of weak solutions to quasilinear elliptic and parabolic equations with controlled growth . Comm. Partial Differential Equations 36(2011), no. 10, 17501777. https://doi.org/10.1080/03605302.2011.571746.
[17] Duzaar, F. and Mingione, G., Gradient estimates via linear and nonlinear potentials . J. Funct. Anal. 259(2010), no. 11, 29612998. https://doi.org/10.1016/j.jfa.2010.08.006.
[18] Duzaar, F. and Mingione, G., Gradient estimates via nonlinear potentials . Amer. J. Math. 133(2011), no. 4, 10931149. https://doi.org/10.1353/ajm.2011.0023.
[19] Gehring, F. W., The L p -integrability of the partial derivatives of a quasiconformal mapping . Acta Math. 130(1973), 265277. https://doi.org/10.1007/BF02392268.
[20] Giaquinta, M. and Modica, G., Regularity results for some classes of higher order nonlinear elliptic systems . J. Reine Angew. Math. 311/312(1979), 145169.
[21] Giaquinta, M. and Struwe, M., On the partial regularity of weak solutions of nonlinear parabolic systems . Math. Z. 179(1982), no. 4, 437451. https://doi.org/10.1007/BF01215058.
[22] Gilbarg, D. and Trudinger, N., Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.
[23] Giusti, E., Direct methods in the calculus of variations. World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
[24] Grafakos, L., Classical Fourier analysis, Second ed., Graduate Texts in Mathematics, 249, Springer, New York, 2008.
[25] Han, Q. and Lin, F., Elliptic partial differential equations, Courant Lecture Notes in Mathematics, 1, Courant Institute of Mathematical Sciences, New York American Mathematical Society, Providence, RI, 1997.
[26] Hardy, G. H., Littlewood, J. E., and Polya, G., Inequalities. Cambridge University Press, Cambridge, 1952.
[27] Hoang, L., Nguyen, T., and Phan, T., Gradient estimates and global existence of smooth solutions to a cross-diffusion system . SIAM J. Math. Anal. 47(2015), no. 3, 21222177. https://doi.org/10.1137/140981447.
[28] Hunt, R. A., On L (p, q) spaces . Einsegnement Math. 12(1966), 249276.
[29] Kim, D., Global regularity of solutions to quasilinear conormal derivative problem with controlled growth . J. Korean Math. Soc. 49(2012), no. 6, 12731299. https://doi.org/10.4134/JKMS.2012.49.6.1273.
[30] Kinnunen, J. and Zhou, S., A local estimate for nonlinear equations with discontinuous coefficients . Comm. Partial Differential Equations 24(1999), no. 11–12, 20432068. https://doi.org/10.1080/03605309908821494.
[31] Ladyzhenskaya, O., Solonikov, V., and Uralt́seva, N., Linear and quasilinear elliptic equations of parabolic type, Translation of Mathematical Monographs, 23, American Mathematical Society, Providence, RI, 1968.
[32] Le, D., Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy . Trans. Amer. Math. Soc. 365(2013), no. 5, 27232753.
[33] Lieberman, G., Second order parabolic differential equations. World Scientific Publishing Co, River Edge, NJ, 2005.
[34] Maugeri, A., Palagachev, D., and Softova, L., Elliptic and parabolic equations with discontinuous coefficients, Mathematical Research, 109, Wiley-VCH Verlag, Berlin, GmbH, Berlin, 2000.
[35] Meyers, N. G., An L p -estimate for the gradient of solutions of second order elliptic divergence equations . Ann. Sculoa Norm. Super. Pisa Cl. Sci. (3) 17(1963), 189206.
[36] Meyers, N. G. and Elcrat, A., Some results on regularity for solutions of nonlinear elliptic systems and quasi-regular functions . Duke Math. J. 42(1975), 121136. https://doi.org/10.1215/S0012-7094-75-04211-8.
[37] Nguyen, Q.-H., Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data . Calc. Var. Partial Differential Equations 54(2015), no. 4, 39273948. https://doi.org/10.1007/s00526-015-0926-y.
[38] Nguyen, T., Interior Calderón-Zygmund estimates for solutions to general parabolic equations of p-Laplacian type . Calc. Var. 56(2017), 173. https://doi.org/10.1007/s00526-017-1265-y.
[39] Nguyen, T. and Phan, T., Interior gradient estimates for quasi-linear elliptic equations . Calc. Var. Partial Differential Equations 55(2016), no. 3, Art. 59. https://doi.org/10.1007/s00526-016-0996-5.
[40] Palagachev, D. K. and Softova, L. G., Quasilinear divergence form parabolic equations in Reifenberg flat domains . Discrete Contin. Dyn. Syst. 31(2011), no. 4, 13971410. https://doi.org/10.3934/dcds.2011.31.1397.
[41] Phan, T., Regularity estimates for BMO-weak solutions of quasi-linear elliptic equations with inhomogeneous boundary conditions . Nonlinear Differ. Equ. Appl. 25(2018), 8. https://doi.org/10.1007/s00030-018-0501-2.
[42] Phan, T., Interior gradient estimates for weak solutions of quasi-linear p-Laplacian type equations . Pacific J. Math, to appear.
[43] Phan, T., Local W 1, p -regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts . Electron. J. Differential Equations(2017), no. 75, 22 pp.
[44] Seregin, G., Silvestre, L., Sverak, V., and Zlatos, A., On divergence-free drifts . J. Differential Equations 252(2012), no. 1, 505540. https://doi.org/10.1016/j.jde.2011.08.039.
[45] Trudinger, N. S., Pointwise estimates and quasilinear parabolic equations . Comm. Pure Appl. Math. 21(1968), 205226. https://doi.org/10.1002/cpa.3160210302.
[46] Wang, L., A geometric approach to the Calderón–Zygmund estimates . Acta Math. Sin. (Engl. Ser.) 19(2003), no. 2, 381396. https://doi.org/10.1007/s10114-003-0264-4.
[47] Zhang, Qi S., A strong regularity result for parabolic equations . Comm. Math. Phys. 244(2004), 245260. https://doi.org/10.1007/s00220-003-0974-6.
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Canadian Journal of Mathematics
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