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Introduction to special issue: “Nonlinear partial differential equations in mathematical fluid dynamics”. (English) Zbl 1398.00067
From the text: This special issue on “Nonlinear Partial Differential Equations in Mathematical Fluid Dynamics” is dedicated to Professor Edriss Saleh Titi on the occasion of his sixtieth birthday, which he celebrated in 2017. Edriss Titi’s wide range of research has focused on the development of analytical and computational techniques for investigating nonlinear phenomena. The study of the Euler and the Navier-Stokes equations of incompressible fluid mechanics has a prominent role in his work, but other related nonlinear partial differential equations arising as models in a wide range of applications in nonlinear science and engineering, have been addressed in his extensive record of publications, many of which has been influential and transformative in the field. The applications include, but are not limited to, fluid mechanics, geophysics, turbulence, chemical reactions, nonlinear fiber optics, and control of complex systems.
MSC:
00B15 Collections of articles of miscellaneous specific interest
00B30 Festschriften
35-06 Proceedings, conferences, collections, etc. pertaining to partial differential equations
76-06 Proceedings, conferences, collections, etc. pertaining to fluid mechanics
Biographic References:
Titi, Edriss Saleh
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[1] Constantin, P.; E, W.; Titi, E. S., Onsager’s conjecture on the energy conservation for solutions of euler’s equations, Comm. Math. Phys., 165, 207-209, (1994) · Zbl 0818.35085
[2] P. Isett, A proof of Onsager’s conjecture, 2016, arXiv preprint arXiv:1608.08301. · Zbl 1335.58018
[3] T. Buckmaster, C. De Lellis, L. Szekelyhidi Jr., V. Vicol, Onsager’s conjecture for admissible weak solutions, 2017, arXiv preprint arXiv:1701.08678. · Zbl 07038033
[4] Bardos, C.; Titi, E. S., Onsager’s conjecture for the incompressible Euler equations in bounded domains, Arch. Ration. Mech. Anal., 228, 1, 197-207, (2018) · Zbl 1390.35241
[5] C. Bardos, E.S. Titi, E. Wiedemann, Onsager’s conjecture with physical boundaries and an application to the vanishing viscosity limlit. arXiv:1803.04939 [math.AP]. · Zbl 1420.35203
[6] Cheskidov, A.; Holm, D. D.; Olson, E.; Titi, E. S., On a Leray-\(\alpha\) model of turbulence, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 629-649, (2005) · Zbl 1145.76386
[7] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S., The Camassa-Holm equations and turbulence, Physica D, 133, 1-4, 49-65, (1999) · Zbl 1194.76069
[8] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S., A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11, 8, 2343-2353, (1999) · Zbl 1147.76357
[9] Chen, S.; Foias, C.; Holm, D. D.; Olson, E.; Titi, E. S.; Wynne, S., Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett., 81, 24, 5338-5341, (1998) · Zbl 1042.76525
[10] Foias, C.; Holm, D. D.; Titi, E. S., The Navier-Stokes-alpha model of fluid turbulence. advances in nonlinear mathematics and science, Physica D, 152/153, 505-519, (2001) · Zbl 1037.76022
[11] Foias, C.; Holm, D. D.; Titi, E. S., The three-dimensional viscous Camassa-Holm equations, and their relation to the Navier-Stokes equations and turbulence theory, J. Dynam. Differential Equations, 14, 1-35, (2002) · Zbl 0995.35051
[12] Cao, Y.; Lunasin, E.; Titi, E. S., Global well-posedness of the three-dimensional viscous and inviscid simplified bardina turbulence models, Commun. Math. Sci., 4, 4, 823-848, (2006) · Zbl 1127.35034
[13] Larios, A.; Petersen, M. R.; Titi, E. S.; Wingate, B., A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization, Theor. Comput. Fluid Dyn., 32, 1, 23-34, (2018)
[14] Bardos, C.; Linshiz, J.; Titi, E., Global regularity for a Birkhoff-rott-\(\alpha\) approximation of the dynamics of vortex sheets of the 2D Euler equations, Physica D, 237, 14-17, 1905-1911, (2008) · Zbl 1143.76384
[15] Cao, C.; Holm, D.; Titi, E. S., On the Clark-\(\alpha\) model of turbulence: global regularity and long-time dynamics, J. Turbul., 6, 20, 1-11, (2005)
[16] Ilyin, A.; Lunasin, E.; Titi, E. S., A modified-Leray-\(\alpha\) subgrid scale model of turbulence, Nonlinearity, 19, 879-897, (2006) · Zbl 1106.35050
[17] Linshiz, J.; Titi, E. S., Analytical study of certain magnetohydrodynamic-alpha models, J. Math. Phys., 48, 6, (2007) · Zbl 1144.81378
[18] Mahalov, A.; Titi, E. S.; Leibovich, S., Invariant helical subspaces for the Navier-Stokes equations, Arch. Ration. Mech. Anal., 112, 193-222, (1990) · Zbl 0708.76044
[19] Ettinger, B.; Titi, E. S., Global existence and uniqueness of weak solutions of 3-D Euler equations with helical symmetry in the absence of vorticity stretching, SIAM J. Math. Anal., 41, 1, 269-296, (2009) · Zbl 1303.76006
[20] Bardos, C.; Lopes Filho, M.; Niu, D.; Nussenzveig, H.; Titi, E. S., Stability of viscous and instabilitty of non-viscous, 2D weak solutions of incompressible fluids under 3D perturbations, SIAM J. Math. Anal., 45, 3, 1871-1885, (2013) · Zbl 1291.35207
[21] Gibbon, J.; Titi, E. S., 3D incompressible Euler with a passive scalar: a road to blow up?, J. Nonlinear Sci., 23, 6, 993-1000, (2013) · Zbl 1292.35216
[22] Cao, C.; Titi, E. S., Global well-posedness of the three-dimensional viscous primitive equations of large scale Ocean and atmospheric dynamics, Ann. of Math., 166, 1, 245-267, (2007) · Zbl 1151.35074
[23] Cao, C.; Ibrahim, S.; Nakanishi, K.; Titi, E. S., Finite-time blowup for hte inviscid primitive equations of oceanic and atmospheric dynamics, Comm. Math. Phys., 337, 473-482, (2015) · Zbl 1317.35262
[24] Li, J.; Titi, E. S., A tropical atmosphere model with moisture: global well-posedness and relaxation limit, Nonlinearity, 29, 2674-2714, (2016) · Zbl 1345.35116
[25] Frierson, D. M.W.; Majda, A. J.; Pauluis, O. M., Dynamics of precipitation fronts in the tropical atmosphere: a noval relaxation limit, Commun. Math. Sci., 2, 591-626, (2004) · Zbl 1160.86303
[26] Azouani, A.; Olson, E.; Titi, E. S., Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24, 2, 277-304, (2014) · Zbl 1291.35168
[27] Azouani, A.; Titi, E. S., Feedback control of nonlinear dissipative systems by finite determining parameters - a reaction-diffusion paradigm, Evolut. Equations Control Theory, 3, 4, 579-594, (2014) · Zbl 1304.35715
[28] Albanez, D.; Nussenzveig-Lopes, H.; Titi, E. S., Continuous data assimilation for the three-dimensional Navier-Stokes-\(\alpha\) model, Asymptot. Anal., 97, 1-2, 139-164, (2016) · Zbl 1344.35078
[29] Altaf, M. U.; Titi, E. S.; Gebrael, T.; Knio, O.; Zhao, L.; McCabe, M. F.; Hoteit, I., Downscaling the 2D Bénard convection equations using continuous data assimilation, Comput. Geosci., 21, 3, 393-410, (2017) · Zbl 1383.65088
[30] Bessaih, H.; Olson, E.; Titi, E. S., Continuous assimilation of data with stochastic noise, Nonlinearity, 28, 729-753, (2015) · Zbl 1308.35161
[31] Farhat, A.; Jolly, M. S.; Titi, E. S., Continuous data assimilation for the 2D Bénard convection through velocity measurements alone, Physica D, 303, 59-66, (2015) · Zbl 1364.76053
[32] Farhat, A.; Lunasin, E.; Titi, E. S., Abridged continuous data assimilation for the 2D Navier-Stokes equations utilizing measurements of only one component of the velocity field, J. Math. Fluid Mech., 18, 1, 1-23, (2015) · Zbl 1334.35202
[33] Jolly, M.; Martinez, V.; Titi, E. S., A data assimilation algorithm for the subcritical surface quasi-geostrophic equation, Adv. Nonlinear Stud., 17, 1, 167-192, (2017) · Zbl 1358.35091
[34] Markowich, P.; Titi, E. S.; Trabelsi, S., Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29, 4, 1292-1328, (2016) · Zbl 1339.35246
[35] Mondaini, C.; Titi, E. S., Uniform in time error estimates for the postprocessing Galerkin method applied to a data assimilation algorithm, SIAM J. Numer. Anal., 56, 1, 78-110, (2018) · Zbl 1384.35067
[36] Farhat, A.; Lunasin, E.; Titi, E. S., On the charney conjecture of data assimilation employing temperature measurements alone: the paradigm of 3D planetary geostrophic model, Math. Clim. Weather Forecast., 2, 1, 61-74, (2016) · Zbl 1364.86017
[37] Charney, J.; Halem, J.; Jastrow, M., Use of incomplete historical data to infer the present state of the atmosphere, J. Atmos. Sci., 26, 1160-1163, (1969)
[38] Constantin, P.; Doering, C. R.; Titi, E. S., Rigorous estimates of small scales in turbulent flows, J. Math. Phys., 37, 6152-6156, (1996) · Zbl 0862.35083
[39] Doering, C. R.; Titi, E. S., Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations, Phys. Fluids, 7, 1384-1390, (1995) · Zbl 1023.76513
[40] Foias, C.; Titi, E. S., Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4, 1, 135-153, (1991) · Zbl 0714.34078
[41] Foias, C.; Jolly, M. S.; Kravchenko, R.; Titi, E. S., A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case, Uspekhi Mat. Nauk, 69, 2, 177-200, (2014), Also Russian Mathematica Surveys, 69(2) (2014) 359-381 · Zbl 1301.35108
[42] Foias, C.; Jolly, M. S.; Lithio, D. D.; Titi, E. S., One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations, J. Nonlinear Sci., 27, 1513-1529, (2017) · Zbl 1379.35211
[43] Cockburn, B.; Jones, D. A.; Titi, E. S., Estimating the number of asymptotic degrees of freedom for nonlinear dissipative systems, Math. Comp., 66, 1073-1087, (1997) · Zbl 0866.35091
[44] Foias, C.; Prodi, G., Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension \(2\), Rend. Semin. Mat. Univ. Padova, 39, 1-34, (1967) · Zbl 0176.54103
[45] Foias, C.; Temam, R., Asymptotic numerical analysis for the Navier-Stokes equations, (Barenblatt; Iooss; Joseph, Nonlinear Dynamics and Turbulence, (1983), Pitman Advanced Pub. Prog. Boston) · Zbl 0555.76030
[46] Foias, C.; Temam, R., Determination of the solutions of the Navier-Stokes equations by a set of nodal values, Math. Comp., 43, 117-133, (1984) · Zbl 0563.35058
[47] Holst, M. J.; Titi, E. S., Determining projections and functionals for weak solutions of the Navier-Stokes equations, Contemp. Math., 204, 125-138, (1997) · Zbl 0873.35062
[48] Jones, D. A.; Titi, E. S., Upper bounds on the number of determining modes, nodes and volume elements for the Navier-Stokes equations, Indiana Univ. Math. J., 32, 3, 875-887, (1993) · Zbl 0796.35128
[49] Biswas, A.; Foias, C.; Nicolaenko, B., Existence time for the 3D Navier-Stokes equations in a generalized Gevrey class, Physica D, 376-377, 5-14, (2018) · Zbl 1398.76032
[50] Bilgin, B. A.; Kalantarov, V. K., Existence of an attractor and determining modes for structurally damped nonlinear wave equations, Physica D, 376-377, 15-22, (2018) · Zbl 1398.35009
[51] Chen, G.; Slemrod, M.; Wang, D., Fluids, geometry, and the onset of Navier-Stokes turbulence in three space dimensions, Physica D, 376-377, 23-30, (2018) · Zbl 1398.76060
[52] Chepyzhov, V.; Ilyin, A.; Zelik, S., Vanishing viscosity limit for global attractors for the damped Navier-Stokes system with stress free boundary conditions, Physica D, 376-377, 31-38, (2018) · Zbl 1398.35145
[53] Hu, W.; Wang, Y.; Wu, J.; Xiao, B.; Yuan, J., Partially dissipative 2D Boussinesq equations with Navier type boundary conditions, Physica D, 376-377, 39-48, (2018) · Zbl 1398.35007
[54] Blocher, J.; Martinez, V.; Olson, E., Data assimilation using using noisy time-averaged measurements, Physica D, 376-377, 49-59, (2018) · Zbl 1398.93119
[55] Gibbon, J. D.; Gupta, A.; Pal, N.; Pandit, R., The role of BKM-type theorems in \(3 D\) Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis, Physica D, 376-377, 60-68, (2018) · Zbl 1398.76033
[56] Anbarlooei, H. R.; Cruz, D. O.A.; Ramos, F.; Santos, C. M.M.; Silva Freire, A. P., On the connection between Kolmogorov microscales and friction in pipe flows of viscoplastic fluids, Physica D, 376-377, 69-77, (2018) · Zbl 1398.76008
[57] Seinen, C.; Khouider, B., Improving the Jacobian free Newton-Krylov method for the viscous-plastic sea ice momentum equation, Physica D, 376-377, 78-93, (2018) · Zbl 1398.65218
[58] Bardos, C.; Golse, F.; Nguyen, T. T.; Sentis, R., The Maxwell-Boltzmann approximation for ion kinetic modeling, Physica D, 376-377, 94-107, (2018) · Zbl 1398.35241
[59] Bodova, K.; Haskovec, J.; Markowich, P., Well posedness and maximum entropy approximation for the dynamics of quantitative traits, Physica D, 376-377, 108-120, (2018) · Zbl 1398.35245
[60] Camliyurt, G.; Kukavica, I., On the Lagrangian and Eulerian analyticity for the Euler equations, Physica D, 376-377, 121-130, (2018) · Zbl 1398.76021
[61] Shvydkoy, R.; Tadmor, E., Eulerian dynamics with a commutator forcing III. fractional diffusion of order \(0 < \alpha < 1\), Physica D, 376-377, 131-137, (2018) · Zbl 1398.35157
[62] Crisan, D.; Holm, D. D., Wave breaking for the stochastic Camassa-Holm equation, Physica D, 376-377, 138-143, (2018) · Zbl 1398.35305
[63] Doering, C. R.; Wu, J.; Zhao, K.; Zheng, X., Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Physica D, 376-377, 144-159, (2018) · Zbl 1398.35164
[64] Galochkina, T.; Marion, M.; Volpert, V., Initiation of reaction-diffusion waves of blood coagulation, Physica D, 376-377, 160-170, (2018) · Zbl 1398.35117
[65] Ilyin, A.; Rykov, Y.; Zelik, S., Hyperbolic relaxation of the 2D Navier-Stokes equations in a bounded domain, Physica D, 376-377, 171-179, (2018) · Zbl 1398.35147
[66] Jiu, Q.; Wang, Y.; Xin, Z., Global classical solutions to two-dimensional compressible Navier-Stokes equations with large data in \(\mathbb{R}^2\), Physica D, 376-377, 180-194, (2018) · Zbl 1398.76193
[67] Constantin, P.; Nguyen, H. Q., Local and global strong solutions for SQG in bounded domains, Physica D, 376-377, 195-203, (2018) · Zbl 1398.35006
[68] Cheskidov, A.; Dai, M., Determining modes for the surface quasi-geostrophic equation, Physica D, 376-377, 204-215, (2018) · Zbl 1398.35162
[69] Balci, N.; Isenberg, A. M.; Jolly, M. S., Turbulence in vertically averaged convection, Physica D, 376-377, 216-227, (2018) · Zbl 1398.76064
[70] Bessaih, H.; Garrido-Atienza, M.; Schmalfuß, B., On 3D Navier-Stokes equations: regularization and uniqueness by delays, Physica D, 376-377, 228-237, (2018) · Zbl 1398.76031
[71] Jiu, Q.; Lopes Filho, M.; Niu, D.; Nussenzveig Lopes, H., The limit of vanishing viscosity for the incompressible 3D Navier-Stokes equations with helical symmetry, Physica D, 376-377, 238-246, (2018) · Zbl 1398.35148
[72] Hong, Y.; Jung, C.; Temam, R., Boundary layer analysis for the stochastic nonlinear reaction-diffusion equations, Physica D, 376-377, 247-258, (2018) · Zbl 1398.35119
[73] Fjordholm, U. S.; Wiedemann, E., Statistical solutions and onsager’s conjecture, Physica D, 376-377, 259-265, (2018) · Zbl 1398.35153
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