Renormalized solutions for stochastic <inline-formula><tex-math id="M1">\begin{document} p \end{document}</tex-math></inline-formula>-Laplace equations with <inline-formula><tex-math id="M2">\begin{document} L^1 \end{document}</tex-math></inline-formula>-initial data: The case of multiplicative noise

Previous Article
A shrinking target theorem for ergodic transformations of the unit interval
DCDS Home
This Issue
Next Article
Gibbs measures for hyperbolic attractors defined by densities

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

Niklas Sapountzoglou and Aleksandra Zimmermann ^,

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.

Keywords: Renormalized solutions, multiplicative noise, stochastic $ p $-Laplace equation, integrable random initial data.

Mathematics Subject Classification: 35K92, 60H15.

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. Bauzet, G. 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. Bénilan, L. Boccardo, T. Gallouét, R. Gariepy, M. 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. Blanchard, F. 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. Dareiotis, M. 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. Debussche, M. 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. Farwig, H. 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. Bauzet, G. 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. Bénilan, L. Boccardo, T. Gallouét, R. Gariepy, M. 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. Blanchard, F. 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. Dareiotis, M. 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. Debussche, M. 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. Farwig, H. 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.

[1]	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
[2]	Xiaohui Zhang, Xuping Zhang. Upper semi-continuity of non-autonomous fractional stochastic $ p $-Laplacian equation driven by additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022081
[3]	Harbir Antil, Mahamadi Warma. Optimal control of the coefficient for the regional fractional $p$-Laplace equation: Approximation and convergence. Mathematical Control and Related Fields, 2019, 9 (1) : 1-38. doi: 10.3934/mcrf.2019001
[4]	Justin Forlano. Almost sure global well posedness for the BBM equation with infinite $ L^{2} $ initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 267-318. doi: 10.3934/dcds.2020011
[5]	Tianling Gao, Qiang Liu, Zhiguang Zhang. Fractional $ 1 $-Laplacian evolution equations to remove multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021254
[6]	Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $ p $-Laplacian. Communications on Pure and Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026
[7]	Nicholas J. Kass, Mohammad A. Rammaha. Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1449-1478. doi: 10.3934/cpaa.2018070
[8]	Nikolay Dimitrov, Stepan Tersian. Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 555-567. doi: 10.3934/dcdsb.2019254
[9]	George Baravdish, Yuanji Cheng, Olof Svensson, Freddie Åström. Generalizations of $ p $-Laplace operator for image enhancement: Part 2. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3477-3500. doi: 10.3934/cpaa.2020152
[10]	Fuzhi Li, Dongmei Xu. Asymptotically autonomous dynamics for non-autonomous stochastic $ g $-Navier-Stokes equation with additive noise. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022087
[11]	Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5681-5705. doi: 10.3934/dcdsb.2020376
[12]	Anhui Gu. Weak pullback mean random attractors for non-autonomous $ p $-Laplacian equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3863-3878. doi: 10.3934/dcdsb.2020266
[13]	Xuping Zhang. Pullback random attractors for fractional stochastic $ p $-Laplacian equation with delay and multiplicative noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1695-1724. doi: 10.3934/dcdsb.2021107
[14]	Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $ (L^2,L^\gamma\cap H_0^1) $-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072
[15]	Sebastian J. Schreiber. The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (1) : 633-644. doi: 10.3934/dcdsb.2020374
[16]	Wenqiang Zhao, Yijin Zhang. High-order Wong-Zakai approximations for non-autonomous stochastic $ p $-Laplacian equations on $ \mathbb{R}^N $. Communications on Pure and Applied Analysis, 2021, 20 (1) : 243-280. doi: 10.3934/cpaa.2020265
[17]	Fuzhi Li, Dongmei Xu, Jiali Yu. Regular measurable backward compact random attractor for $ g $-Navier-Stokes equation. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3137-3157. doi: 10.3934/cpaa.2020136
[18]	Jianing Chen, Bixiang Wang. Random attractors of supercritical wave equations driven by infinite-dimensional additive noise on $ \mathbb{R}^n $. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022093
[19]	Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control and Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030
[20]	Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Niklas Sapountzoglou Aleksandra Zimmermann
on Google Scholar
Niklas Sapountzoglou Aleksandra Zimmermann

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Niklas Sapountzoglou Aleksandra Zimmermann
on Google Scholar
Niklas Sapountzoglou Aleksandra Zimmermann

[Back to Top]