American Institute of Mathematical Sciences

Advanced
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

[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

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
[1]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[2]

Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020284
[3]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352
[4]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[5]

Yi Zhou, Jianli Liu. The initial-boundary value problem on a strip for the equation of time-like extremal surfaces. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 381-397. doi: 10.3934/dcds.2009.23.381
[6]

Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021005
[7]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080
[8]

Wenlong Sun, Jiaqi Cheng, Xiaoying Han. Random attractors for 2D stochastic micropolar fluid flows on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 693-716. doi: 10.3934/dcdsb.2020189
[9]

Sebastian J. Schreiber. The $ P^* $ rule in the stochastic Holt-Lawton model of apparent competition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 633-644. doi: 10.3934/dcdsb.2020374
[10]

Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel. Homogeneous darboux polynomials and generalising integrable ODE systems. Journal of Computational Dynamics, 2021, 8 (1) : 1-8. doi: 10.3934/jcd.2021001
[11]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[12]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432
[13]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175
[14]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391
[15]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320
[16]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445
[17]

Fuensanta Andrés, Julio Muñoz, Jesús Rosado. Optimal design problems governed by the nonlocal $ p $-Laplacian equation. Mathematical Control & Related Fields, 2021, 11 (1) : 119-141. doi: 10.3934/mcrf.2020030
[18]

Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020293
[19]

Raffaele Folino, Ramón G. Plaza, Marta Strani. Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020403
[20]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (16)
  • HTML views (96)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences