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doi: 10.3934/dcds.2020367

Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data

Niklas Sapountzoglou and  Aleksandra Zimmermann ,

University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany

* Corresponding author

Received  December 2019 Revised  August 2020 Published  November 2020

We consider a $ p $-Laplace evolution problem with stochastic forcing on a bounded domain $ D\subset\mathbb{R}^d $ with homogeneous Dirichlet boundary conditions for $ 1<p<\infty $. The additive noise term is given by a stochastic integral in the sense of Itô. The technical difficulties arise from the merely integrable random initial data $ u_0 $ under consideration. Due to the poor regularity of the initial data, estimates in $ W^{1,p}_0(D) $ are available with respect to truncations of the solution only and therefore well-posedness results have to be formulated in the sense of generalized solutions. We extend the notion of renormalized solution for this type of SPDEs, show well-posedness in this setting and study the Markov properties of solutions.

Keywords: Stochastic p-Laplace equation, integrable random initial data, renormalized solutions.
Mathematics Subject Classification: 35K92, 60H15.
Citation: Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020367
References:
[1]

S. Attanasio and F. Flandoli, Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplication noise, Comm. Partial Differential Equations, 36 (2011), 1455-1474.  doi: 10.1080/03605302.2011.585681.  Google Scholar

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G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, New York, London, 1979.  Google Scholar

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P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273.   Google Scholar

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D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pures Appl., 77 (1998), 117-151.  doi: 10.1016/S0021-7824(98)80067-6.  Google Scholar

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D. Blanchard and F. Murat, Renormalised solutions of nonlinear parabolic problems with $L^1$ data: Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.  doi: 10.1017/S0308210500026986.  Google Scholar

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D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution of a fairly general class of nonlinear parabolic problems, Journal of Differential Equations, 177 (2001), 331-374.  doi: 10.1006/jdeq.2000.4013.  Google Scholar

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D. Breit, Regularity theory for nonlinear systems of SPDEs, Manuscripta Math., 146 (2015), 329-349.  doi: 10.1007/s00229-014-0704-8.  Google Scholar

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D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter, Berlin, 2018.  Google Scholar

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P. Catuogno and C. Olivera, $L^p$-solutions of the stochastic transport equation, Random Oper. Stoch. Equ., 21 (2013), 125-134.  doi: 10.1515/rose-2013-0007.  Google Scholar

[12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. 2. Edition, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
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J. I. Diaz and F. de Thélin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal., 25 (1994), 1085-1111.  doi: 10.1137/S0036141091217731.  Google Scholar

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R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

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E. Feireisl, Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford 2004.  Google Scholar

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D. Fellah and É. Pardoux, Une formule d'Itô dans des espaces de Banach, et Application, In: Körezlioǧlu H., Üstünel A.S. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol. 31. Birkhäuser, Boston, MA, 1992.  Google Scholar

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B. Gess and M. Hofmanová, Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE, Ann. Probab., 46 (2018), 2495-2544.  doi: 10.1214/17-AOP1231.  Google Scholar

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M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), e6, 75 pp. doi: 10.1017/fmp.2015.2.  Google Scholar

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M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.  Google Scholar

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L. Hörmander, The Analyis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis., Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

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T. Komorowski, S. Peszat and T. Szarek, On ergodicity of some Markov processes, Ann. Probab., 38 (2010), 1401-1443.  doi: 10.1214/09-AOP513.  Google Scholar

[23]

N. V. Krylov and B. L. Rozovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Stochastic evolution equations, J. Soviet Math., 16 (1981), 1233-1277.   Google Scholar

[24]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

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M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 1-63.  doi: 10.4064/dm426-0-1.  Google Scholar

[26]

E. Pardoux, Equations aux Dérivées Partielles Stochastiques non Linéaires Monotones, University of Paris, 1975. PhD-thesis. Google Scholar

[27]

S. Punshon-Smith and S. Smith, On the Boltzmann equation with stochastic kinetic transport: Global existence of renormalized martingale solutions, Arch. Rational Mech. Anal., 229 (2018), 627-708.  doi: 10.1007/s00205-018-1225-5.  Google Scholar

[28]

G. Vallet and A. Zimmermann, Well-posedness for a pseudomonotone evolution problem with multiplicative noise, J. Evol. Equ., 19 (2019), 153-202.  doi: 10.1007/s00028-018-0472-0.  Google Scholar

[29] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar

show all references

References:
[1]

S. Attanasio and F. Flandoli, Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplication noise, Comm. Partial Differential Equations, 36 (2011), 1455-1474.  doi: 10.1080/03605302.2011.585681.  Google Scholar

[2]

P. Baldi, Stochastic Calculus, An Introduction Through Theory and Exercises. Universitext, Springer, 2017. doi: 10.1007/978-3-319-62226-2.  Google Scholar

[3]

G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, New York, London, 1979.  Google Scholar

[4]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273.   Google Scholar

[5]

D. Blanchard and H. Redwane, Renormalized solutions for a class of nonlinear evolution problems, J. Math. Pures Appl., 77 (1998), 117-151.  doi: 10.1016/S0021-7824(98)80067-6.  Google Scholar

[6]

D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Anal., 21 (1993), 725-743.  doi: 10.1016/0362-546X(93)90120-H.  Google Scholar

[7]

D. Blanchard and F. Murat, Renormalised solutions of nonlinear parabolic problems with $L^1$ data: Existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.  doi: 10.1017/S0308210500026986.  Google Scholar

[8]

D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution of a fairly general class of nonlinear parabolic problems, Journal of Differential Equations, 177 (2001), 331-374.  doi: 10.1006/jdeq.2000.4013.  Google Scholar

[9]

D. Breit, Regularity theory for nonlinear systems of SPDEs, Manuscripta Math., 146 (2015), 329-349.  doi: 10.1007/s00229-014-0704-8.  Google Scholar

[10]

D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter, Berlin, 2018.  Google Scholar

[11]

P. Catuogno and C. Olivera, $L^p$-solutions of the stochastic transport equation, Random Oper. Stoch. Equ., 21 (2013), 125-134.  doi: 10.1515/rose-2013-0007.  Google Scholar

[12] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. 2. Edition, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.  Google Scholar
[13]

B. Delamotte, A hint of renormalization, Am. J. Phys., 72 (2004), 170-184.   Google Scholar

[14]

J. I. Diaz and F. de Thélin, On a nonlinear parabolic problem arising in some models related to turbulent flows, SIAM J. Math. Anal., 25 (1994), 1085-1111.  doi: 10.1137/S0036141091217731.  Google Scholar

[15]

R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[16]

E. Feireisl, Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford 2004.  Google Scholar

[17]

D. Fellah and É. Pardoux, Une formule d'Itô dans des espaces de Banach, et Application, In: Körezlioǧlu H., Üstünel A.S. (eds) Stochastic Analysis and Related Topics. Progress in Probability, vol. 31. Birkhäuser, Boston, MA, 1992.  Google Scholar

[18]

B. Gess and M. Hofmanová, Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE, Ann. Probab., 46 (2018), 2495-2544.  doi: 10.1214/17-AOP1231.  Google Scholar

[19]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), e6, 75 pp. doi: 10.1017/fmp.2015.2.  Google Scholar

[20]

M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504.  doi: 10.1007/s00222-014-0505-4.  Google Scholar

[21]

L. Hörmander, The Analyis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis., Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-642-61497-2.  Google Scholar

[22]

T. Komorowski, S. Peszat and T. Szarek, On ergodicity of some Markov processes, Ann. Probab., 38 (2010), 1401-1443.  doi: 10.1214/09-AOP513.  Google Scholar

[23]

N. V. Krylov and B. L. Rozovski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$, Stochastic evolution equations, J. Soviet Math., 16 (1981), 1233-1277.   Google Scholar

[24]

W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.  Google Scholar

[25]

M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. (Rozprawy Mat.), 426 (2004), 1-63.  doi: 10.4064/dm426-0-1.  Google Scholar

[26]

E. Pardoux, Equations aux Dérivées Partielles Stochastiques non Linéaires Monotones, University of Paris, 1975. PhD-thesis. Google Scholar

[27]

S. Punshon-Smith and S. Smith, On the Boltzmann equation with stochastic kinetic transport: Global existence of renormalized martingale solutions, Arch. Rational Mech. Anal., 229 (2018), 627-708.  doi: 10.1007/s00205-018-1225-5.  Google Scholar

[28]

G. Vallet and A. Zimmermann, Well-posedness for a pseudomonotone evolution problem with multiplicative noise, J. Evol. Equ., 19 (2019), 153-202.  doi: 10.1007/s00028-018-0472-0.  Google Scholar

[29] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
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