August  2022, 42(8): 3979-4002. doi: 10.3934/dcds.2022041

Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise

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

* Corresponding author

Received  March 2021 Revised  March 2022 Published  August 2022 Early access  April 2022

We consider a $ p $-Laplace evolution problem with multiplicative noise on a bounded domain $ D\subset\mathbb{R}^d $ with homogeneous Dirichlet boundary conditions for $ 1<p<\infty $. The random initial data is merely integrable. Consequently, the key estimates are available with respect to truncations of the solution. We introduce the notion of renormalized solutions for multiplicative stochastic $ p $-Laplace equations with $ L^1 $-initial data and study existence and uniqueness of solutions in this framework.

Citation: Niklas Sapountzoglou, Aleksandra Zimmermann. Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3979-4002. doi: 10.3934/dcds.2022041
References:
[1]

S. Attanasio and F. Flandoli, Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative 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]

C. BauzetG. Vallet and P. Wittbold, The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperbolic Differ. Equ., 9 (2012), 661-709.  doi: 10.1142/S0219891612500221.

[4]

P. BénilanL. BoccardoT. GallouétR. GariepyM. Pierre and J. 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]

I. H. Biswas and A. K. Majee, Stochastic conservation laws: Weak-in-time formulation and strong entropy condition, J. Funct. Anal., 267 (2014), 2199-2252.  doi: 10.1016/j.jfa.2014.07.008.

[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. BlanchardF. Murat and H. Redwane, Existence and uniqueness of a renormalized solution of a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.  doi: 10.1006/jdeq.2000.4013.

[9]

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.

[10]

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

[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, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[13]

K. DareiotisM. Gerencsér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differ. Equ., 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[14]

A. DebusscheM. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab., 44 (2016), 1916-1955.  doi: 10.1214/15-AOP1013.

[15]

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

[16]

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.

[17]

R. FarwigH. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005), 21-53.  doi: 10.1007/BF02588049.

[18]

B. Fehrman and B. Gess, Well-posedness of stochastic porous media equations with nonlinear, conservative noise, Arch. Ration. Mech. Anal., 233 (2019), 249-322.  doi: 10.1007/s00205-019-01357-w.

[19] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. 
[20]

D. Fellah and É. Pardoux, Une formule d'Itô dans des espaces de Banach, et application, Stochastic Analysis and Related Topics, Progress in Probability, 31 (1992), 197-209. 

[21]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlinear Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.

[22]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), 6, 75 pp. doi: 10.1017/fmp.2015.2.

[23]

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

[24]

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. 

[25]

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

[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]

T. Roubìcek, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.

[29]

N. Sapountzoglou and A. Zimmermann, Renormalized solutions for a stochastic $p$-Laplace equation with $L^1$ initial data, Proceedings of the Fifteenth International Conference Zaragoza-Pau on Mathematics and its Applications, Monogr. Mat. García Galdeano, 42 (2020).

[30]

N. Sapountzoglou and A. Zimmermann, Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data, Discrete Contin. Dyn. Syst., 41 (2021), 2341-2376.  doi: 10.3934/dcds.2020367.

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[32]

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.

show all references

References:
[1]

S. Attanasio and F. Flandoli, Renormalized solutions for stochastic transport equations and the regularization by bilinear multiplicative 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]

C. BauzetG. Vallet and P. Wittbold, The Cauchy problem for conservation laws with a multiplicative stochastic perturbation, J. Hyperbolic Differ. Equ., 9 (2012), 661-709.  doi: 10.1142/S0219891612500221.

[4]

P. BénilanL. BoccardoT. GallouétR. GariepyM. Pierre and J. 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]

I. H. Biswas and A. K. Majee, Stochastic conservation laws: Weak-in-time formulation and strong entropy condition, J. Funct. Anal., 267 (2014), 2199-2252.  doi: 10.1016/j.jfa.2014.07.008.

[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. BlanchardF. Murat and H. Redwane, Existence and uniqueness of a renormalized solution of a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.  doi: 10.1006/jdeq.2000.4013.

[9]

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.

[10]

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

[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, Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9780511666223.
[13]

K. DareiotisM. Gerencsér and B. Gess, Entropy solutions for stochastic porous media equations, J. Differ. Equ., 266 (2019), 3732-3763.  doi: 10.1016/j.jde.2018.09.012.

[14]

A. DebusscheM. Hofmanová and J. Vovelle, Degenerate parabolic stochastic partial differential equations: Quasilinear case, Ann. Probab., 44 (2016), 1916-1955.  doi: 10.1214/15-AOP1013.

[15]

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

[16]

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.

[17]

R. FarwigH. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005), 21-53.  doi: 10.1007/BF02588049.

[18]

B. Fehrman and B. Gess, Well-posedness of stochastic porous media equations with nonlinear, conservative noise, Arch. Ration. Mech. Anal., 233 (2019), 249-322.  doi: 10.1007/s00205-019-01357-w.

[19] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford Lecture Series in Mathematics and its Applications, 26. Oxford University Press, Oxford, 2004. 
[20]

D. Fellah and É. Pardoux, Une formule d'Itô dans des espaces de Banach, et application, Stochastic Analysis and Related Topics, Progress in Probability, 31 (1992), 197-209. 

[21]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlinear Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974.

[22]

M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum Math. Pi, 3 (2015), 6, 75 pp. doi: 10.1017/fmp.2015.2.

[23]

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

[24]

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. 

[25]

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

[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]

T. Roubìcek, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2005.

[29]

N. Sapountzoglou and A. Zimmermann, Renormalized solutions for a stochastic $p$-Laplace equation with $L^1$ initial data, Proceedings of the Fifteenth International Conference Zaragoza-Pau on Mathematics and its Applications, Monogr. Mat. García Galdeano, 42 (2020).

[30]

N. Sapountzoglou and A. Zimmermann, Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data, Discrete Contin. Dyn. Syst., 41 (2021), 2341-2376.  doi: 10.3934/dcds.2020367.

[31]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[32]

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.

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