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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 & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042
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show all references

References:
[1]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 25-60. doi: 10.1016/S0294-1449(03)00031-3.  Google Scholar

[2]

Elsevier/Academic Press, Amsterdam, 2003.  Google Scholar

[3]

Glas. Mat. Ser. III, 35 (2000), 161-177.  Google Scholar

[4]

H. Amann, Anisotropic function spaces on singular manifolds,, preprint http://arxiv.org/pdf/1204.0606.pdf., ().  doi: 10.1002/mana.201100157.  Google Scholar

[5]

Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102.  Google Scholar

[6]

J. Math. Fluid Mech., 16 (2014), 225-241. doi: 10.1007/s00021-013-0149-y.  Google Scholar

[7]

J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2.  Google Scholar

[8]

J. Math. Fluid Mech., 11 (2009), 258-273. doi: 10.1007/s00021-008-0258-1.  Google Scholar

[9]

Nonlinear Anal., 75 (2012), 4346-4354. doi: 10.1016/j.na.2012.03.021.  Google Scholar

[10]

Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 1173-1191. doi: 10.3934/dcdss.2013.6.1173.  Google Scholar

[11]

J. Math. Fluid Mech., 13 (2011), 387-404. doi: 10.1007/s00021-010-0025-y.  Google Scholar

[12]

Duke Math J., 8 (1961), 301-324.  Google Scholar

[13]

Academic Press, Inc., Boston, 1988.  Google Scholar

[14]

Springer-Verlag, Berlin, Heiderberg, New York, 1976.  Google Scholar

[15]

Dokl. Akad. Nauk SSSR, 126 (1959), 1163-1165.  Google Scholar

[16]

Nauka, Moscow, 1975.  Google Scholar

[17]

J. Reine Angew. Math., 357 (1985), 1-22. doi: 10.1515/crll.1985.357.1.  Google Scholar

[18]

Mat. Sb. (N. S.), 43 (1957), 451-503.  Google Scholar

[19]

Mem. Amer. Math. Soc., 221 (2013), no. 1041. doi: 10.1090/S0065-9266-2012-00664-2.  Google Scholar

[20]

Math. Methods Appl. Sci., 33 (2010), 1995-2010. doi: 10.1002/mma.1314.  Google Scholar

[21]

PhD Thesis, Warsaw 2015, (available at http://mmns.mimuw.edu.pl/phd/Burczak_phd.pdf, accessed 24.06.2016). Google Scholar

[22]

manuscripta math., 144 (2014), 51-90. doi: 10.1007/s00229-013-0640-z.  Google Scholar

[23]

J. Burczak and P. Kaplický, Interior regularity of space derivatives to an evolutionary, symmetric $\varphi$-Laplacian,, preprint: arXiv:1507.05843 [math.AP], ().   Google Scholar

[24]

J. Math. Fluid Mech., 10 (2008), 455-487. doi: 10.1007/s00021-008-0282-1.  Google Scholar

[25]

Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[26]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 705-751. doi: 10.1016/j.anihpc.2004.10.011.  Google Scholar

[27]

Memoirs A.M.S. 214, 2011. doi: 10.1090/S0065-9266-2011-00614-3.  Google Scholar

[28]

Ann. Scuola Norm. Sup. Pisa., 9 (2010), 1-46.  Google Scholar

[29]

SIAM J. Math. Anal., 47 (2015), 3917-3943. doi: 10.1137/141000725.  Google Scholar

[30]

Comput. Math. Appl., 53 (2007), 509-520. doi: 10.1016/j.camwa.2006.02.039.  Google Scholar

[31]

Trudy Mat. Inst. Steklova, 66 (1962), 364-383.  Google Scholar

[32]

J. Math. Pures Appl., 45 (1966), 143-206.  Google Scholar

[33]

Int. J. Numer. Anal. Model. Ser. B, 4 (2013), 215-223.  Google Scholar

[34]

Academia, Prague, 1977.  Google Scholar

[35]

Mat. Vest., 24 (1972), 289-303.  Google Scholar

[36]

Z. Anal. Anwendungen, 24 (2005), 467-486. doi: 10.4171/ZAA/1251.  Google Scholar

[37]

J. Math. Fluid Mech., 10 (2008), 71-88. doi: 10.1007/s00021-006-0217-7.  Google Scholar

[38]

NoDEA, 9 (2002), 175-195. doi: 10.1007/s00030-002-8123-z.  Google Scholar

[39]

Adv. Differential Equations, 6 (2001), 257-302.  Google Scholar

[40]

J. Math Pures Appl., 9 (1927), 337-425. Google Scholar

[41]

Publ. Res. Inst. Math. Sci., 9 (1974), 325-396.  Google Scholar

[42]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 18 (1991), 1-11.  Google Scholar

[43]

Doklady Akad. Nauk SSSR (N.S.), 76 (1951), 785-788.  Google Scholar

[44]

Electron. J. Diff. Equ., 10 (2011), 1-10.  Google Scholar

[45]

volume 1748 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2000. doi: 10.1007/BFb0104029.  Google Scholar

[46]

Math. Nachr., 285 (2012), 1082-1106. doi: 10.1002/mana.201100011.  Google Scholar

[47]

Ann. Mat. Pura Appl., 157 (1990), 117-148. doi: 10.1007/BF01765315.  Google Scholar

[48]

Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9193-6.  Google Scholar

[49]

Ann. Mat. Pura Appl., 134 (1983), 241-266. doi: 10.1007/BF01773507.  Google Scholar

[50]

Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[51]

Birkhäuser Verlag, Basel, 2006.  Google Scholar

[52]

Acta Math., 138 (1977), 219-240.  Google Scholar

[53]

Springer Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.  Google Scholar

[54]

Duke Math. J., 12 (1945), 47-76.  Google Scholar

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