Well-posedness of renormalized solutions for a stochastic <inline-formula><tex-math id="M1">\begin{document} p \end{document}</tex-math></inline-formula>-Laplace equation with <inline-formula><tex-math id="M2">\begin{document} L^1 \end{document}</tex-math></inline-formula>-initial data

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May 2021, 41(5): 2341-2376. 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 May 2021 Early access 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 and Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. 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.
[2]	P. Baldi, Stochastic Calculus, An Introduction Through Theory and Exercises. Universitext, Springer, 2017. doi: 10.1007/978-3-319-62226-2.
[3]	G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, New York, London, 1979.
[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.
[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.
[6]	D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Anal., 21 (1993), 725-743. doi: 10.1016/0362-546X(93)90120-H.
[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.
[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.
[9]	D. Breit, Regularity theory for nonlinear systems of SPDEs, Manuscripta Math., 146 (2015), 329-349. doi: 10.1007/s00229-014-0704-8.
[10]	D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter, Berlin, 2018.
[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.
[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.
[13]	B. Delamotte, A hint of renormalization, Am. J. Phys., 72 (2004), 170-184.
[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.
[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.
[16]	E. Feireisl, Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford 2004.
[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.
[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.
[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.
[20]	M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504. doi: 10.1007/s00222-014-0505-4.
[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.
[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.
[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.
[24]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.
[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.
[26]	E. Pardoux, Equations aux Dérivées Partielles Stochastiques non Linéaires Monotones, University of Paris, 1975. PhD-thesis.
[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.
[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.
[29]	J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

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.
[2]	P. Baldi, Stochastic Calculus, An Introduction Through Theory and Exercises. Universitext, Springer, 2017. doi: 10.1007/978-3-319-62226-2.
[3]	G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, New York, London, 1979.
[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.
[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.
[6]	D. Blanchard, Truncations and monotonicity methods for parabolic equations, Nonlinear Anal., 21 (1993), 725-743. doi: 10.1016/0362-546X(93)90120-H.
[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.
[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.
[9]	D. Breit, Regularity theory for nonlinear systems of SPDEs, Manuscripta Math., 146 (2015), 329-349. doi: 10.1007/s00229-014-0704-8.
[10]	D. Breit, E. Feireisl and M. Hofmanová, Stochastically Forced Compressible Fluid Flows, De Gruyter, Berlin, 2018.
[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.
[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.
[13]	B. Delamotte, A hint of renormalization, Am. J. Phys., 72 (2004), 170-184.
[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.
[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.
[16]	E. Feireisl, Dynamics of Viscous Compressible Fluids. Volume 26 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford 2004.
[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.
[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.
[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.
[20]	M. Hairer, A theory of regularity structures, Invent. Math., 198 (2014), 269-504. doi: 10.1007/s00222-014-0505-4.
[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.
[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.
[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.
[24]	W. Liu and M. Röckner, Stochastic Partial Differential Equations: An Introduction, Universitext, Springer, Cham, 2015. doi: 10.1007/978-3-319-22354-4.
[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.
[26]	E. Pardoux, Equations aux Dérivées Partielles Stochastiques non Linéaires Monotones, University of Paris, 1975. PhD-thesis.
[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.
[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.
[29]	J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987. doi: 10.1017/CBO9781139171755.

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