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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 |
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.
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. |
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