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March  2020, 19(3): 1697-1745. doi: 10.3934/cpaa.2020069

Global higher integrability of weak solutions of porous medium systems

Kristian Moring 1, , Christoph Scheven 2,, , Sebastian Schwarzacher 3, and  Thomas Singer 4,

1. 

Department of Mathematics and Systems Analysis, Aalto University, P. O. Box 11100, FI-00076 Aalto, Finland
2. 

Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann-Str. 9, 45127 Essen, Germany
3. 

Katedra matematické analyźy, Matematicko-fyzikální fakulta Univerzity Karlovy, Sokolovská 83,186 75 Praha 8, Czech Republic
4. 

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany

* Corresponding author

Received  March 2019 Revised  August 2019 Published  November 2019

Fund Project: K. Moring has been supported by the Magnus Ehrnrooth foundation. T. Singer has been supported by the DFG-Project SI 2464/1-1 ``Highly nonlinear evolutionary problems".

We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by
$ \begin{equation*} \partial_t u-\Delta(|u|^{m-1}u) = \mathrm{div}\,F\,, \end{equation*} $
where
$ m>1 $
. More precisely, we prove that under suitable assumptions the spatial gradient
$ D(|u|^{m-1}u) $
 of any weak solution is integrable to a larger power than the natural power
$ 2 $
. Our analysis includes both the case of the lateral boundary and the initial boundary.
Keywords: Porous medium type systems, higher integrability, gradient estimates.
Mathematics Subject Classification: 35B65, 35K65, 35K40, 35K55.
Citation: Kristian Moring, Christoph Scheven, Sebastian Schwarzacher, Thomas Singer. Global higher integrability of weak solutions of porous medium systems. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1697-1745. doi: 10.3934/cpaa.2020069
References:
[1]

K. Adimurthi and S. Byun, Boundary Higher Integrability for Very Weak Solutions of Quasilinear Parabolic Equations, J. Math. Pures Appl., 121 (2019), 244-285.  doi: 10.1016/j.matpur.2018.06.005.  Google Scholar

[2]

V. Bögelein, Higher integrability for weak solutions of higher order degenerate parabolic systems, Ann. Acad. Sci. Fenn. Math., 33 (2008), 387-412.   Google Scholar

[3]

V. Bögelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher integrability for doubly nonlinear parabolic systems, Preprint., Google Scholar

[4]

V. Bögelein, F. Duzaar, R. Korte and C. Scheven, The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8 (2019), 1004-1034.  doi: 10.1515/anona-2017-0270.  Google Scholar

[5]

V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p, q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.  doi: 10.1007/s00205-013-0646-4.  Google Scholar

[6]

V. Bögelein, F. Duzaar and C. Scheven, Higher integrability for the singular porous medium system, Preprint, 2018. Google Scholar

[7]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21-54.  doi: 10.1007/s00030-009-0038-5.  Google Scholar

[8]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Universitytext xv, 387, New York, NY, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[9]

E. DiBenedetto and A. Friedman, Hölder estimates for non-linear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1–22. 1985. doi: 10.1515/crll.1985.357.1.  Google Scholar

[10]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, 2011. doi: 10.1007/978-1-4614-1584-8.  Google Scholar

[11]

L. Diening, P. Kaplický and S. Schwarzacher, BMO estimates for the $p$-Laplacian, Nonlinear Anal., 75 (2012), 637-650.  doi: 10.1016/j.na.2011.08.065.  Google Scholar

[12]

L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. doi: 10.1007/978-1-4614-1584-8.  Google Scholar

[13]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969.  Google Scholar

[14]

F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.  doi: 10.1007/BF02392268.  Google Scholar

[15]

U. Gianazza and S. Schwarzacher, Self-improving property of degenerate parabolic equations of porous medium-type, Amer. J. Math., 141 (2019), 399-446.  doi: 10.1353/ajm.2019.0009.  Google Scholar

[16]

U. Gianazza and S. Schwarzacher, Self-improving property of the fast diffusion equation, J. Funct. Anal., 277 (2019), 108291. doi: 10.1016/j.jfa.2019.108291.  Google Scholar

[17] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983.   Google Scholar
[18]

M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Ang. Math., 311/312 (1979), 145-169.   Google Scholar

[19]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 185-208.   Google Scholar

[20]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar

[21]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Company, Tuck Link, Singapore, 2003. doi: 10.1142/9789812795557.  Google Scholar

[22]

P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar

[23]

L. I. Hedberg, Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81.  doi: 10.1007/BF02385982.  Google Scholar

[24] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, New York, 1993.   Google Scholar
[25]

T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909.  doi: 10.1016/0362-546X(94)90127-9.  Google Scholar

[26]

J. Kinnunen and J. L. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J., 102 (2000), 253-271.  doi: 10.1215/S0012-7094-00-10223-2.  Google Scholar

[27]

J. Kinnunen and J. Lewis, Very weak solutions of parabolic systems of p-Laplacian type, Ark. Mat., 40 (2002), 105-132.  doi: 10.1007/BF02384505.  Google Scholar

[28]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435.  doi: 10.1007/s10231-005-0160-x.  Google Scholar

[29]

J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations, Adv. Cal. Var., 3 (2010), 29-48.  doi: 10.1515/ACV.2010.002.  Google Scholar

[30]

J. L. Lewis, Uniformly fat sets, Trans. Am. Math. Soc., 308 (1988), 177-196.  doi: 10.2307/2000957.  Google Scholar

[31]

N. G. Meyers and A. Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136.   Google Scholar

[32]

M. Parviainen, Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl., 188 (2009), 333-358.  doi: 10.1007/s10231-008-0079-0.  Google Scholar

[33]

M. Parviainen, Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations, 246 (2009), 512-540.  doi: 10.1016/j.jde.2008.06.013.  Google Scholar

[34]

S. Schwarzacher, Hölder-Zygmund estimates for degenerate parabolic systems, J. Differential Equations, 256 (2014), 2423-2448.  doi: 10.1016/j.jde.2014.01.009.  Google Scholar

show all references

References:
[1]

K. Adimurthi and S. Byun, Boundary Higher Integrability for Very Weak Solutions of Quasilinear Parabolic Equations, J. Math. Pures Appl., 121 (2019), 244-285.  doi: 10.1016/j.matpur.2018.06.005.  Google Scholar

[2]

V. Bögelein, Higher integrability for weak solutions of higher order degenerate parabolic systems, Ann. Acad. Sci. Fenn. Math., 33 (2008), 387-412.   Google Scholar

[3]

V. Bögelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher integrability for doubly nonlinear parabolic systems, Preprint., Google Scholar

[4]

V. Bögelein, F. Duzaar, R. Korte and C. Scheven, The higher integrability of weak solutions of porous medium systems, Adv. Nonlinear Anal., 8 (2019), 1004-1034.  doi: 10.1515/anona-2017-0270.  Google Scholar

[5]

V. Bögelein, F. Duzaar and P. Marcellini, Parabolic systems with $p, q$-growth: a variational approach, Arch. Ration. Mech. Anal., 210 (2013), 219-267.  doi: 10.1007/s00205-013-0646-4.  Google Scholar

[6]

V. Bögelein, F. Duzaar and C. Scheven, Higher integrability for the singular porous medium system, Preprint, 2018. Google Scholar

[7]

V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21-54.  doi: 10.1007/s00030-009-0038-5.  Google Scholar

[8]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, Universitytext xv, 387, New York, NY, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[9]

E. DiBenedetto and A. Friedman, Hölder estimates for non-linear degenerate parabolic systems, J. Reine Angew. Math., 357 (1985), 1–22. 1985. doi: 10.1515/crll.1985.357.1.  Google Scholar

[10]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations, Springer Monographs in Mathematics, 2011. doi: 10.1007/978-1-4614-1584-8.  Google Scholar

[11]

L. Diening, P. Kaplický and S. Schwarzacher, BMO estimates for the $p$-Laplacian, Nonlinear Anal., 75 (2012), 637-650.  doi: 10.1016/j.na.2011.08.065.  Google Scholar

[12]

L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. doi: 10.1007/978-1-4614-1584-8.  Google Scholar

[13]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag, New York, 1969.  Google Scholar

[14]

F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 130 (1973), 265-277.  doi: 10.1007/BF02392268.  Google Scholar

[15]

U. Gianazza and S. Schwarzacher, Self-improving property of degenerate parabolic equations of porous medium-type, Amer. J. Math., 141 (2019), 399-446.  doi: 10.1353/ajm.2019.0009.  Google Scholar

[16]

U. Gianazza and S. Schwarzacher, Self-improving property of the fast diffusion equation, J. Funct. Anal., 277 (2019), 108291. doi: 10.1016/j.jfa.2019.108291.  Google Scholar

[17] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, 1983.   Google Scholar
[18]

M. Giaquinta and G. Modica, Regularity results for some classes of higher order nonlinear elliptic systems, J. Reine Ang. Math., 311/312 (1979), 145-169.   Google Scholar

[19]

M. Giaquinta and G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 3 (1986), 185-208.   Google Scholar

[20]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar

[21]

E. Giusti, Direct Methods in the Calculus of Variations, World Scientific Publishing Company, Tuck Link, Singapore, 2003. doi: 10.1142/9789812795557.  Google Scholar

[22]

P. Hajłasz, P. Koskela and H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal., 254 (2008), 1217-1234.  doi: 10.1016/j.jfa.2007.11.020.  Google Scholar

[23]

L. I. Hedberg, Two approximation problems in function spaces, Ark. Mat., 16 (1978), 51-81.  doi: 10.1007/BF02385982.  Google Scholar

[24] J. Heinonen, T. Kilpeläinen and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs, Oxford University Press, New York, 1993.   Google Scholar
[25]

T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909.  doi: 10.1016/0362-546X(94)90127-9.  Google Scholar

[26]

J. Kinnunen and J. L. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type, Duke Math. J., 102 (2000), 253-271.  doi: 10.1215/S0012-7094-00-10223-2.  Google Scholar

[27]

J. Kinnunen and J. Lewis, Very weak solutions of parabolic systems of p-Laplacian type, Ark. Mat., 40 (2002), 105-132.  doi: 10.1007/BF02384505.  Google Scholar

[28]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435.  doi: 10.1007/s10231-005-0160-x.  Google Scholar

[29]

J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations, Adv. Cal. Var., 3 (2010), 29-48.  doi: 10.1515/ACV.2010.002.  Google Scholar

[30]

J. L. Lewis, Uniformly fat sets, Trans. Am. Math. Soc., 308 (1988), 177-196.  doi: 10.2307/2000957.  Google Scholar

[31]

N. G. Meyers and A. Elcrat, Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions, Duke Math. J., 42 (1975), 121-136.   Google Scholar

[32]

M. Parviainen, Global gradient estimates for degenerate parabolic equations in nonsmooth domains, Ann. Mat. Pura Appl., 188 (2009), 333-358.  doi: 10.1007/s10231-008-0079-0.  Google Scholar

[33]

M. Parviainen, Reverse Hölder inequalities for singular parabolic equations near the boundary, J. Differential Equations, 246 (2009), 512-540.  doi: 10.1016/j.jde.2008.06.013.  Google Scholar

[34]

S. Schwarzacher, Hölder-Zygmund estimates for degenerate parabolic systems, J. Differential Equations, 256 (2014), 2423-2448.  doi: 10.1016/j.jde.2014.01.009.  Google Scholar

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