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On the asymptotic properties for stationary solutions to the Navier-Stokes equations
Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data
University of Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany |
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.
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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. |
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P. Baldi, Stochastic Calculus, An Introduction Through Theory and Exercises. Universitext, Springer, 2017.
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G. I. Barenblatt, Similarity, Self-Similarity, and Intermediate Asymptotics, New York, London, 1979. |
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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.
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[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.
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M. Hairer,
A theory of regularity structures, Invent. Math., 198 (2014), 269-504.
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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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