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November  2016, 15(6): 2401-2445. doi: 10.3934/cpaa.2016042

## Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences

 1 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-656 Warsaw, Poland 2 Charles University, Faculty of Mathematics and Physics, Department of Mathematical Analysis, Sokolovská 83, 186 75 Prague 8

Received  March 2016 Revised  July 2016 Published  September 2016

We consider an evolutionary, non-degenerate, symmetric $p$-Laplacian. By symmetric we mean that the full gradient of $p$-Laplacian is replaced by its symmetric part, which causes a breakdown of the Uhlenbeck structure. We derive interior regularity of time derivatives of its local weak solution. To circumvent the space-time growth mismatch, we devise a new local regularity technique of iterations in Nikolskii-Bochner spaces. It is interesting by itself, as it may be modified to provide new regularity results for the full-gradient $p$-Laplacian case with lower-order dependencies. Finally, having our regularity result for time derivatives, we obtain respective regularity of the main part. The Appendix on Nikolskii-Bochner spaces, that includes theorems on their embeddings and interpolations, may be of independent interest.
Citation: Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042
##### References:
 [1] E. Acerbi, G. Mingione and G. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 25-60. doi: 10.1016/S0294-1449(03)00031-3. [2] R. Adams and J. Fournier, Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003. [3] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III, 35 (2000), 161-177. [4] H. Amann, Anisotropic function spaces on singular manifolds, preprint http://arxiv.org/pdf/1204.0606.pdf. doi: 10.1002/mana.201100157. [5] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102. [6] H. Bae and B. Jin, Regularity of non-Newtonian fluids, J. Math. Fluid Mech., 16 (2014), 225-241. doi: 10.1007/s00021-013-0149-y. [7] H. Beirão da Veiga, Navier-Stokes equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2. [8] H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 258-273. doi: 10.1007/s00021-008-0258-1. [9] H. Beirão da Veiga and F. Crispo, On the global $W^{2,q}$ regularity for nonlinear $N$-systems of the $p$-Laplacian type in $n$ space variables, Nonlinear Anal., 75 (2012), 4346-4354. doi: 10.1016/j.na.2012.03.021. [10] H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173-1191. doi: 10.3934/dcdss.2013.6.1173. [11] H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404. doi: 10.1007/s00021-010-0025-y. [12] A. Benedek and R. Panzone, The spaces $L^p$ with mixed norm, Duke Math J., 8 (1961), 301-324. [13] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, 1988. [14] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin, Heiderberg, New York, 1976. [15] O. Besov, On some families of functional spaces. Imbedding and extension theorems (Russian), Dokl. Akad. Nauk SSSR, 126 (1959), 1163-1165. [16] O. Besov, V. Il'in and S. Nikol'skii, Integral Representations of Functions, and Embedding Theorems (Russian), Nauka, Moscow, 1975. [17] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. [18] B. Bojarski, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients (Russian), Mat. Sb. (N. S.), 43 (1957), 451-503. [19] V. Bögelein, F. Duzaar and G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Mem. Amer. Math. Soc., 221 (2013), no. 1041. doi: 10.1090/S0065-9266-2012-00664-2. [20] M. Bulíček, F. Ettwein, P. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids Math. Methods Appl. Sci., 33 (2010), 1995-2010. doi: 10.1002/mma.1314. [21] J. Burczak, Regularity of solutions to nonlinear non-diagonal evolutionary systems, PhD Thesis, Warsaw 2015, (available at http://mmns.mimuw.edu.pl/phd/Burczak_phd.pdf, accessed 24.06.2016). [22] J. Burczak, Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems, manuscripta math., 144 (2014), 51-90. doi: 10.1007/s00229-013-0640-z. [23] J. Burczak and P. Kaplický, Interior regularity of space derivatives to an evolutionary, symmetric $\varphi$-Laplacian, preprint: arXiv:1507.05843 [math.AP] [24] F. Crispo and C. Grisanti, On the existence, uniqueness and $C^{1, \gamma} (\O) \cap W^{2,2} (\O)$ regularity for a class of shear-thinning fluids, J. Math. Fluid Mech., 10 (2008), 455-487. doi: 10.1007/s00021-008-0282-1. [25] E. DiBenedetto, Degenerate Parabolic Systems, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [26] F. Duzaar and G. Mingione, Second order parabolic systems, optimal regularity and singular sets of solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 705-751. doi: 10.1016/j.anihpc.2004.10.011. [27] F. Duzaar, G. Mingione and K. Steffen, Parabolic Systems with Polynomial Growth and Regularity, Memoirs A.M.S. 214, 2011. doi: 10.1090/S0065-9266-2011-00614-3. [28] L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. [29] J. Frehse and S. Schwarzacher, On regularity of the time derivative for degenerate parabolic systems, SIAM J. Math. Anal., 47 (2015), 3917-3943. doi: 10.1137/141000725. [30] M. Fuchs and G. Seregin, Global nonlinear evolution problem for generalized Newtonian fluids: local initial regularity of the strong solution, Comput. Math. Appl., 53 (2007), 509-520. doi: 10.1016/j.camwa.2006.02.039. [31] K. Golovkin, On equivalent normalizations of fractional spaces (Russian), Trudy Mat. Inst. Steklova, 66 (1962), 364-383. [32] P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures Appl., 45 (1966), 143-206. [33] B. Jin, On the Caccioppoli inequality of the unsteady Stokes system, Int. J. Numer. Anal. Model. Ser. B, 4 (2013), 215-223. [34] O. John, A. Kufner and S. Fučík, Function Spaces, Academia, Prague, 1977. [35] H. Johnen, Inequalities connected with the moduli of smoothness, Mat. Vest., 24 (1972), 289-303. [36] P. Kaplický, Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions, Z. Anal. Anwendungen, 24 (2005), 467-486. doi: 10.4171/ZAA/1251. [37] P. Kaplický, Regularity of flow of anisotropic fluid, J. Math. Fluid Mech., 10 (2008), 71-88. doi: 10.1007/s00021-006-0217-7. [38] P. Kaplický, J. Málek and J Stará, Global-in-time Hoder continuity of the velocity gradients for fluids with shear-dependent viscosities, NoDEA, 9 (2002), 175-195. doi: 10.1007/s00030-002-8123-z. [39] J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case $p \ge 2$, Adv. Differential Equations, 6 (2001), 257-302. [40] A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, J. Math Pures Appl., 9 (1927), 337-425. [41] T. Muramatu, On Besov spaces and Sobolev spaces of generalized functions definded on a general region, Publ. Res. Inst. Math. Sci., 9 (1974), 325-396. [42] J. Nečas and V. Sverák, On regularity of solutions of nonlinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1991), 1-11. [43] S. Nikol'skii, Some inequalities for entire functions of finite degree of several variables and their application (Russian), Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 785-788. [44] P. Rabier, Vector-valued Morrey's embedding theorem and Hölder continuity in parabolic problems, Electron. J. Diff. Equ., 10 (2011), 1-10. [45] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, volume 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. [46] B. Scharf, H. Schmeisser and W. Sickel, Traces of vector-valued Sobolev spaces, Math. Nachr., 285 (2012), 1082-1106. doi: 10.1002/mana.201100011. [47] J. Simon, Sobolev, Besov and Nikol'skii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl., 157 (1990), 117-148. doi: 10.1007/BF01765315. [48] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6. [49] P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134 (1983), 241-266. doi: 10.1007/BF01773507. [50] H. Triebel, Theory of Function Spaces I, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. [51] H. Triebel, Theory of Function Spaces III, Birkhäuser Verlag, Basel, 2006. [52] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. [53] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. [54] A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76.

show all references

##### References:
 [1] E. Acerbi, G. Mingione and G. Seregin, Regularity results for parabolic systems related to a class of non-Newtonian fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 25-60. doi: 10.1016/S0294-1449(03)00031-3. [2] R. Adams and J. Fournier, Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003. [3] H. Amann, Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat. Ser. III, 35 (2000), 161-177. [4] H. Amann, Anisotropic function spaces on singular manifolds, preprint http://arxiv.org/pdf/1204.0606.pdf. doi: 10.1002/mana.201100157. [5] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102. [6] H. Bae and B. Jin, Regularity of non-Newtonian fluids, J. Math. Fluid Mech., 16 (2014), 225-241. doi: 10.1007/s00021-013-0149-y. [7] H. Beirão da Veiga, Navier-Stokes equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2. [8] H. Beirão da Veiga, Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 258-273. doi: 10.1007/s00021-008-0258-1. [9] H. Beirão da Veiga and F. Crispo, On the global $W^{2,q}$ regularity for nonlinear $N$-systems of the $p$-Laplacian type in $n$ space variables, Nonlinear Anal., 75 (2012), 4346-4354. doi: 10.1016/j.na.2012.03.021. [10] H. Beirão da Veiga and F. Crispo, On the global regularity for nonlinear systems of the $p$-Laplacian type, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173-1191. doi: 10.3934/dcdss.2013.6.1173. [11] H. Beirão da Veiga, P. Kaplický and M. Růžička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404. doi: 10.1007/s00021-010-0025-y. [12] A. Benedek and R. Panzone, The spaces $L^p$ with mixed norm, Duke Math J., 8 (1961), 301-324. [13] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc., Boston, 1988. [14] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Springer-Verlag, Berlin, Heiderberg, New York, 1976. [15] O. Besov, On some families of functional spaces. Imbedding and extension theorems (Russian), Dokl. Akad. Nauk SSSR, 126 (1959), 1163-1165. [16] O. Besov, V. Il'in and S. Nikol'skii, Integral Representations of Functions, and Embedding Theorems (Russian), Nauka, Moscow, 1975. [17] E. DiBenedetto and A. Friedman, Hölder estimates for nonlinear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1. [18] B. Bojarski, Generalized solutions of a system of differential equations of first order and of elliptic type with discontinuous coefficients (Russian), Mat. Sb. (N. S.), 43 (1957), 451-503. [19] V. Bögelein, F. Duzaar and G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Mem. Amer. Math. Soc., 221 (2013), no. 1041. doi: 10.1090/S0065-9266-2012-00664-2. [20] M. Bulíček, F. Ettwein, P. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids Math. Methods Appl. Sci., 33 (2010), 1995-2010. doi: 10.1002/mma.1314. [21] J. Burczak, Regularity of solutions to nonlinear non-diagonal evolutionary systems, PhD Thesis, Warsaw 2015, (available at http://mmns.mimuw.edu.pl/phd/Burczak_phd.pdf, accessed 24.06.2016). [22] J. Burczak, Almost everywhere Hölder continuity of gradients to non-diagonal parabolic systems, manuscripta math., 144 (2014), 51-90. doi: 10.1007/s00229-013-0640-z. [23] J. Burczak and P. Kaplický, Interior regularity of space derivatives to an evolutionary, symmetric $\varphi$-Laplacian, preprint: arXiv:1507.05843 [math.AP] [24] F. Crispo and C. Grisanti, On the existence, uniqueness and $C^{1, \gamma} (\O) \cap W^{2,2} (\O)$ regularity for a class of shear-thinning fluids, J. Math. Fluid Mech., 10 (2008), 455-487. doi: 10.1007/s00021-008-0282-1. [25] E. DiBenedetto, Degenerate Parabolic Systems, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2. [26] F. Duzaar and G. Mingione, Second order parabolic systems, optimal regularity and singular sets of solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 705-751. doi: 10.1016/j.anihpc.2004.10.011. [27] F. Duzaar, G. Mingione and K. Steffen, Parabolic Systems with Polynomial Growth and Regularity, Memoirs A.M.S. 214, 2011. doi: 10.1090/S0065-9266-2011-00614-3. [28] L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motion of generalized Newtonian fluids, Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46. [29] J. Frehse and S. Schwarzacher, On regularity of the time derivative for degenerate parabolic systems, SIAM J. Math. Anal., 47 (2015), 3917-3943. doi: 10.1137/141000725. [30] M. Fuchs and G. Seregin, Global nonlinear evolution problem for generalized Newtonian fluids: local initial regularity of the strong solution, Comput. Math. Appl., 53 (2007), 509-520. doi: 10.1016/j.camwa.2006.02.039. [31] K. Golovkin, On equivalent normalizations of fractional spaces (Russian), Trudy Mat. Inst. Steklova, 66 (1962), 364-383. [32] P. Grisvard, Commutativité de deux foncteurs d'interpolation et applications, J. Math. Pures Appl., 45 (1966), 143-206. [33] B. Jin, On the Caccioppoli inequality of the unsteady Stokes system, Int. J. Numer. Anal. Model. Ser. B, 4 (2013), 215-223. [34] O. John, A. Kufner and S. Fučík, Function Spaces, Academia, Prague, 1977. [35] H. Johnen, Inequalities connected with the moduli of smoothness, Mat. Vest., 24 (1972), 289-303. [36] P. Kaplický, Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions, Z. Anal. Anwendungen, 24 (2005), 467-486. doi: 10.4171/ZAA/1251. [37] P. Kaplický, Regularity of flow of anisotropic fluid, J. Math. Fluid Mech., 10 (2008), 71-88. doi: 10.1007/s00021-006-0217-7. [38] P. Kaplický, J. Málek and J Stará, Global-in-time Hoder continuity of the velocity gradients for fluids with shear-dependent viscosities, NoDEA, 9 (2002), 175-195. doi: 10.1007/s00030-002-8123-z. [39] J. Málek, J. Nečas and M. Růžička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case $p \ge 2$, Adv. Differential Equations, 6 (2001), 257-302. [40] A. Marchaud, Sur les dérivées et sur les différences des fonctions de variables réelles, J. Math Pures Appl., 9 (1927), 337-425. [41] T. Muramatu, On Besov spaces and Sobolev spaces of generalized functions definded on a general region, Publ. Res. Inst. Math. Sci., 9 (1974), 325-396. [42] J. Nečas and V. Sverák, On regularity of solutions of nonlinear parabolic systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1991), 1-11. [43] S. Nikol'skii, Some inequalities for entire functions of finite degree of several variables and their application (Russian), Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 785-788. [44] P. Rabier, Vector-valued Morrey's embedding theorem and Hölder continuity in parabolic problems, Electron. J. Diff. Equ., 10 (2011), 1-10. [45] M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, volume 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029. [46] B. Scharf, H. Schmeisser and W. Sickel, Traces of vector-valued Sobolev spaces, Math. Nachr., 285 (2012), 1082-1106. doi: 10.1002/mana.201100011. [47] J. Simon, Sobolev, Besov and Nikol'skii fractional spaces: imbeddings and comparisons for vector valued spaces on an interval, Ann. Mat. Pura Appl., 157 (1990), 117-148. doi: 10.1007/BF01765315. [48] L. Simon, Theorems on Regularity and Singularity of Energy Minimizing Maps, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6. [49] P. Tolksdorf, Everywhere-regularity for some quasilinear systems with a lack of ellipticity, Ann. Mat. Pura Appl., 134 (1983), 241-266. doi: 10.1007/BF01773507. [50] H. Triebel, Theory of Function Spaces I, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1. [51] H. Triebel, Theory of Function Spaces III, Birkhäuser Verlag, Basel, 2006. [52] K. Uhlenbeck, Regularity for a class of non-linear elliptic systems, Acta Math., 138 (1977), 219-240. [53] E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0. [54] A. Zygmund, Smooth functions, Duke Math. J., 12 (1945), 47-76.
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