January  2017, 16(1): 189-208. doi: 10.3934/cpaa.2017009

Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D

Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA

Received  March 2016 Revised  July 2016 Published  November 2016

We show the continuity of a specific cost functional $J(\phi) =\mathbb{E} \sup_{ t \in [0, T]}(\varphi(\mathcal{L}[t, u_\phi(t), \phi(t)]))$ of the SNSE in 2D on an open bounded nonperiodic domain $\mathcal{O}$ with respect to a special set of feedback controls $\{\phi_n\}_{n \geq 0}$, where $\varphi(x) =\log(1 + x)^{1-\epsilon}$ with $0 < \epsilon < 1$.

Citation: Kerem Uǧurlu. Continuity of cost functional and optimal feedback controls for the stochastic Navier Stokes equation in 2D. Communications on Pure and Applied Analysis, 2017, 16 (1) : 189-208. doi: 10.3934/cpaa.2017009
References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput.Fluid Dynamics., 1 (1990), 303-325.

[2]

A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.

[3]

H. Breckner, Existence of optimal and epsilon-optimal controls for the stochastic NavierStokes Equation, Nonlinear Analysis, 51 (2002), 95-118 doi: 10.1016/S0362-546X(01)00814-8.

[4]

J. Bricmont, A. Kupiainen and R. Lefevere, Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations, J. Statist. Phys., 100 (2000), 743-756. doi: 10.1023/A:1018627609718.

[5]

Z. Brzézniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations, Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , vol. 52, R. Neth. Acad. Arts Sci. , Amsterdam, 2000, 85-98.

[6]

M. Capinski and NJ. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore, 1995. doi: 10.1142/9789812831958.

[7]

M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with application to NavierStokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140.

[8]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200. doi: 10.1007/PL00001415.

[9]

H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), 509-543. doi: 10.1017/S0022112093001880.

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.

[11]

A. B. Cruzeiro, Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes, Exposition. Math., 7 (1989), 73-82.

[12]

G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947. doi: 10.1016/S0021-7824(03)00025-4.

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[14]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, (2011), to appear. doi: 10.1016/j.physd.2011.03.009.

[15]

R. Durrett, Probability: Theory and Examples, Cambridge University Press, Cambridge, 2013.

[16]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM Series Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088.

[17]

F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1942, Springer Berlin / Heidelberg, 2008, 51-150. doi: 10.1007/978-3-540-78493-7_2.

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.

[19]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241 (electronic). doi: 10.1090/S0002-9947-02-02975-6.

[20]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[21]

A. V. Fursikov and MJ. Vishik, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988.

[22]

D. Gatarek, Existence of optimal controls for stochastic evolution systems, in: G. Da Prato et al. , Control of Partial Differential Equations, IFIP WG 7. 2 Conference, Villa Madruzzo, Trento, Italy, January 4-9,1993, New York, Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. , 165 (1994), 81-86.

[23]

D. Gatarek and J. Sobczyk, On the existence of optimal controls of Hilbert space-valued diffusions, SIAM Control Optim., 32 (1994), 170-175. doi: 10.1137/S0363012992226260.

[24]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability, 42 (2014), 80-145. doi: 10.1214/12-AOP773.

[25]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.

[26]

W. Grecksch, Stochastische Evolutionsgleichungen undderen Steuerung, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987.

[27]

S. B. Kuksin, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2006. doi: 10.4171/021.

[28]

I. Kukavica, K. Uğurlu and M. Ziane, On the Galerkin approximation and norm estimates of the stochastic Navier-Stokes equations with multiplicative noise, submitted.

[29]

I. Kukavica and V. Vicol, On moments for strong solutions of the 2D stochastic Navier-Stokes equations in a bounded domain, Asymptotic Analysis, 90 (2014), 189-206.

[30]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity, J. Statist. Phys. , 108 (2002), 1157-1179, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. doi: 10.1023/A:1019799700126.

[31]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310. doi: 10.1137/S0036141002409167.

[32]

R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176. doi: 10.1214/009117904000000630.

[33]

J.-L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.

[34]

C. Odasso, Spatial smoothness of the stationary solutions of the 3D Navier-Stokes equations, Electron. J. Probab., 11 (2006), 686-699. doi: 10.1214/EJP.v11-336.

[35]

G. Da Prato and A. Debussche, Control of the stochastic burgers model of turbulence, SIAM J. Control Optim., 37 (1999), 1123-1149. doi: 10.1137/S0363012996311307.

[36]

G. Da Prato and A. Ichikawa, Stability and quadratic control for linear stochastic equations with unbounded coefficients, Boll. Unione Mat. Ital., Ⅵ. Ser. B, 4 (1985), 987-1001.

[37]

C. Prévôt and M. Röckner, A Concise Course On Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.

[38]

A. Shirikyan, Analyticity of solutions and Kolmogorov's dissipation scale for 2D NavierStokes equations, Evolution equations (Warsaw, 2001), Banach Center Publ. , vol. 60, Polish Acad. Sci. , Warsaw, 2003, 49-53.

[39]

S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyper viscosities, Appl. Math. Optim., 41 (2000), 255-308. doi: 10.1007/s0024599110140.

[40]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979.

[41]

C. Tudor, Optimal control for semi-linear evolution equations, Appl. Math. Optim., 20 (1989), 319-331. doi: 10.1007/BF01447659.

[42]

C. Tudor, Optimal and optimal control for the stochastic linear-quadratic problem, Math. Nachr., 145 (1990), 135-149. doi: 10.1002/mana.19901450111.

[43]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thèse, Université Pierre et Marie Curie, Paris, 1976.

[44]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Ⅲ: Variational Methods and Optimization, Springer, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

show all references

References:
[1]

F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput.Fluid Dynamics., 1 (1990), 303-325.

[2]

A. Bensoussan and R. Temam, Équations stochastiques du type Navier-Stokes, J. Functional Analysis, 13 (1973), 195-222.

[3]

H. Breckner, Existence of optimal and epsilon-optimal controls for the stochastic NavierStokes Equation, Nonlinear Analysis, 51 (2002), 95-118 doi: 10.1016/S0362-546X(01)00814-8.

[4]

J. Bricmont, A. Kupiainen and R. Lefevere, Probabilistic estimates for the two-dimensional stochastic Navier-Stokes equations, J. Statist. Phys., 100 (2000), 743-756. doi: 10.1023/A:1018627609718.

[5]

Z. Brzézniak and S. Peszat, Strong local and global solutions for stochastic Navier-Stokes equations, Infinite dimensional stochastic analysis (Amsterdam, 1999), Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet. , vol. 52, R. Neth. Acad. Arts Sci. , Amsterdam, 2000, 85-98.

[6]

M. Capinski and NJ. Cutland, Nonstandard Methods for Stochastic Fluid Mechanics, World Scientific, Singapore, 1995. doi: 10.1142/9789812831958.

[7]

M. Capiński and D. Gatarek, Stochastic equations in Hilbert space with application to NavierStokes equations in any dimension, J. Funct. Anal., 126 (1994), 26-35. doi: 10.1006/jfan.1994.1140.

[8]

M. Capiński and S. Peszat, Local existence and uniqueness of strong solutions to 3-D stochastic Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., 4 (1997), 185-200. doi: 10.1007/PL00001415.

[9]

H. Choi, R. Temam, P. Moin and J. Kim, Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253 (1993), 509-543. doi: 10.1017/S0022112093001880.

[10]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1988.

[11]

A. B. Cruzeiro, Solutions et mesures invariantes pour des équations d'évolution stochastiques du type Navier-Stokes, Exposition. Math., 7 (1989), 73-82.

[12]

G. Da Prato and A. Debussche, Ergodicity for the 3D stochastic Navier-Stokes equations, J. Math. Pures Appl., 82 (2003), 877-947. doi: 10.1016/S0021-7824(03)00025-4.

[13]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511666223.

[14]

A. Debussche, N. Glatt-Holtz and R. Temam, Local martingale and pathwise solutions for an abstract fluids model, Physica D, (2011), to appear. doi: 10.1016/j.physd.2011.03.009.

[15]

R. Durrett, Probability: Theory and Examples, Cambridge University Press, Cambridge, 2013.

[16]

I. Ekeland and R. Temam, Convex Analysis and Variational Problems, SIAM Series Classics in Applied Mathematics, SIAM, Philadelphia, 1999. doi: 10.1137/1.9781611971088.

[17]

F. Flandoli, An introduction to 3d stochastic fluid dynamics, SPDE in Hydrodynamic: Recent Progress and Prospects, Lecture Notes in Mathematics, vol. 1942, Springer Berlin / Heidelberg, 2008, 51-150. doi: 10.1007/978-3-540-78493-7_2.

[18]

F. Flandoli and D. Gatarek, Martingale and stationary solutions for stochastic Navier-Stokes equations, Probab. Theory Related Fields, 102 (1995), 367-391. doi: 10.1007/BF01192467.

[19]

F. Flandoli and M. Romito, Partial regularity for the stochastic Navier-Stokes equations, Trans. Amer. Math. Soc., 354 (2002), 2207-2241 (electronic). doi: 10.1090/S0002-9947-02-02975-6.

[20]

C. Foias and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova, 39 (1967), 1-34.

[21]

A. V. Fursikov and MJ. Vishik, Mathematical Problems in Statistical Hydromechanics, Kluwer, Dordrecht, 1988.

[22]

D. Gatarek, Existence of optimal controls for stochastic evolution systems, in: G. Da Prato et al. , Control of Partial Differential Equations, IFIP WG 7. 2 Conference, Villa Madruzzo, Trento, Italy, January 4-9,1993, New York, Marcel Dekker, Inc. Lect. Notes Pure Appl. Math. , 165 (1994), 81-86.

[23]

D. Gatarek and J. Sobczyk, On the existence of optimal controls of Hilbert space-valued diffusions, SIAM Control Optim., 32 (1994), 170-175. doi: 10.1137/S0363012992226260.

[24]

N. Glatt-Holtz and V. Vicol, Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise, The Annals of Probability, 42 (2014), 80-145. doi: 10.1214/12-AOP773.

[25]

N. Glatt-Holtz and M. Ziane, Strong pathwise solutions of the stochastic Navier-Stokes system, Advances in Differential Equations, 14 (2009), 567-600.

[26]

W. Grecksch, Stochastische Evolutionsgleichungen undderen Steuerung, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987.

[27]

S. B. Kuksin, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2006. doi: 10.4171/021.

[28]

I. Kukavica, K. Uğurlu and M. Ziane, On the Galerkin approximation and norm estimates of the stochastic Navier-Stokes equations with multiplicative noise, submitted.

[29]

I. Kukavica and V. Vicol, On moments for strong solutions of the 2D stochastic Navier-Stokes equations in a bounded domain, Asymptotic Analysis, 90 (2014), 189-206.

[30]

J. C. Mattingly, The dissipative scale of the stochastics Navier-Stokes equation: regularization and analyticity, J. Statist. Phys. , 108 (2002), 1157-1179, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. doi: 10.1023/A:1019799700126.

[31]

R. Mikulevicius and B. L. Rozovskii, Stochastic Navier-Stokes equations for turbulent flows, SIAM J. Math. Anal., 35 (2004), 1250-1310. doi: 10.1137/S0036141002409167.

[32]

R. Mikulevicius and B. L. Rozovskii, Global L2-solutions of stochastic Navier-Stokes equations, Ann. Probab., 33 (2005), 137-176. doi: 10.1214/009117904000000630.

[33]

J.-L. Menaldi and S. S. Sritharan, Stochastic 2-D Navier-Stokes equation, Applied Mathematics and Optimization, 46 (2002), 31-53. doi: 10.1007/s00245-002-0734-6.

[34]

C. Odasso, Spatial smoothness of the stationary solutions of the 3D Navier-Stokes equations, Electron. J. Probab., 11 (2006), 686-699. doi: 10.1214/EJP.v11-336.

[35]

G. Da Prato and A. Debussche, Control of the stochastic burgers model of turbulence, SIAM J. Control Optim., 37 (1999), 1123-1149. doi: 10.1137/S0363012996311307.

[36]

G. Da Prato and A. Ichikawa, Stability and quadratic control for linear stochastic equations with unbounded coefficients, Boll. Unione Mat. Ital., Ⅵ. Ser. B, 4 (1985), 987-1001.

[37]

C. Prévôt and M. Röckner, A Concise Course On Stochastic Partial Differential Equations, Lecture Notes in Mathematics, vol. 1905, Springer, Berlin, 2007.

[38]

A. Shirikyan, Analyticity of solutions and Kolmogorov's dissipation scale for 2D NavierStokes equations, Evolution equations (Warsaw, 2001), Banach Center Publ. , vol. 60, Polish Acad. Sci. , Warsaw, 2003, 49-53.

[39]

S. S. Sritharan, Deterministic and stochastic control of Navier-Stokes equation with linear, monotone, and hyper viscosities, Appl. Math. Optim., 41 (2000), 255-308. doi: 10.1007/s0024599110140.

[40]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin, 1979.

[41]

C. Tudor, Optimal control for semi-linear evolution equations, Appl. Math. Optim., 20 (1989), 319-331. doi: 10.1007/BF01447659.

[42]

C. Tudor, Optimal and optimal control for the stochastic linear-quadratic problem, Math. Nachr., 145 (1990), 135-149. doi: 10.1002/mana.19901450111.

[43]

M. Viot, Solutions faibles d'équations aux dérivées partielles non linéaires, Thèse, Université Pierre et Marie Curie, Paris, 1976.

[44]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. Ⅲ: Variational Methods and Optimization, Springer, New York, 1985. doi: 10.1007/978-1-4612-5020-3.

[1]

Ana Bela Cruzeiro. Navier-Stokes and stochastic Navier-Stokes equations via Lagrange multipliers. Journal of Geometric Mechanics, 2019, 11 (4) : 553-560. doi: 10.3934/jgm.2019027

[2]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[3]

Enrique Fernández-Cara. Motivation, analysis and control of the variable density Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (6) : 1021-1090. doi: 10.3934/dcdss.2012.5.1021

[4]

Yuri Bakhtin. Lyapunov exponents for stochastic differential equations with infinite memory and application to stochastic Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 697-709. doi: 10.3934/dcdsb.2006.6.697

[5]

Hakima Bessaih, Benedetta Ferrario. Statistical properties of stochastic 2D Navier-Stokes equations from linear models. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2927-2947. doi: 10.3934/dcdsb.2016080

[6]

Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209

[7]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations and Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355

[8]

Takeshi Taniguchi. The existence and decay estimates of the solutions to $3$D stochastic Navier-Stokes equations with additive noise in an exterior domain. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4323-4341. doi: 10.3934/dcds.2014.34.4323

[9]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123

[10]

Tongtong Liang. The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022127

[11]

Pavel I. Plotnikov, Jan Sokolowski. Optimal shape control of airfoil in compressible gas flow governed by Navier-Stokes equations. Evolution Equations and Control Theory, 2013, 2 (3) : 495-516. doi: 10.3934/eect.2013.2.495

[12]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations and Control Theory, 2022, 11 (2) : 347-371. doi: 10.3934/eect.2020110

[13]

Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control and Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61

[14]

Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602

[15]

Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149

[16]

Anna Amirdjanova, Jie Xiong. Large deviation principle for a stochastic navier-Stokes equation in its vorticity form for a two-dimensional incompressible flow. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651

[17]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5421-5448. doi: 10.3934/dcdsb.2020352

[18]

Manil T. Mohan, Sivaguru S. Sritharan. $\mathbb{L}^p-$solutions of the stochastic Navier-Stokes equations subject to Lévy noise with $\mathbb{L}^m(\mathbb{R}^m)$ initial data. Evolution Equations and Control Theory, 2017, 6 (3) : 409-425. doi: 10.3934/eect.2017021

[19]

Gung-Min Gie, Makram Hamouda, Roger Temam. Asymptotic analysis of the Navier-Stokes equations in a curved domain with a non-characteristic boundary. Networks and Heterogeneous Media, 2012, 7 (4) : 741-766. doi: 10.3934/nhm.2012.7.741

[20]

Cheng Wang. Convergence analysis of Fourier pseudo-spectral schemes for three-dimensional incompressible Navier-Stokes equations. Electronic Research Archive, 2021, 29 (5) : 2915-2944. doi: 10.3934/era.2021019

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (151)
  • HTML views (147)
  • Cited by (0)

Other articles
by authors

[Back to Top]