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Vectorial variational problems in \(L^\infty\) constrained by the Navier-Stokes equations. (English) Zbl 1481.35307


MSC:

35Q30 Navier-Stokes equations
35D35 Strong solutions to PDEs
35A15 Variational methods applied to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
49J40 Variational inequalities
49K20 Optimality conditions for problems involving partial differential equations
49K35 Optimality conditions for minimax problems
49M41 PDE constrained optimization (numerical aspects)
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[1] Amann, H., Compact embeddings of vector-valued Sobolev and Besov spaces, Glas. Mat., 35, 161-177 (2000) · Zbl 0997.46029
[2] Amann, H., On the strong solvability of the Navier-Stokes equations, J. Math. Fluid Mech., 2, 16-98 (2000) · Zbl 0989.35107
[3] Ansini, N.; Prinari, F., On the lower semicontinuity of supremal functional under differential constraints, ESAIM Control Optim. Calc. Var., 21, 1053-1075 (2015) · Zbl 1336.49015
[4] Aronsson, G., Minimization problems for the functional \(####\), Ark. Mat., 6, 33-53 (1965) · Zbl 0156.12502
[5] Aronsson, G., Minimization problems for the functional \(####\). (II), Ark. Mat., 6, 409-431 (1966)
[6] Aronsson, G.; Barron, E. N., L^∞ variational problems with running costs and constraints, Appl. Math. Optim., 65, 53-90 (2012) · Zbl 1242.49049
[7] Ayanbayev, B.; Katzourakis, N., Vectorial variational principles in L^∞ and their characterisation through PDE systems, Appl. Math. Optim., 83, 833-838 (2019) · Zbl 1465.35174
[8] Ayanbayev, B.; Katzourakis, N., A pointwise characterisation of the PDE system of vectorial calculus of variations in L^∞, Proc. R. Soc. Edinburgh A, 150, 1653-1669 (2019) · Zbl 1444.35058
[9] Barron, E. N.; Bocea, M.; Jensen, R. R., Viscosity solutions of stationary Hamilton-Jacobi equations and minimizers of L^∞ functionals, Proc. Am. Math. Soc., 145, 5257-5265 (2017) · Zbl 1380.35051
[10] Barron, E. N.; Jensen, R., Minimizing the L^∞ norm of the gradient with an energy constraint, Commun. Partial. Differ. Equ., 30, 1741-1772 (2005) · Zbl 1105.35028
[11] Barron, E. N.; Jensen, R. R.; Wang, C. Y., The Euler equation and absolute minimizers of L^∞ functionals, Arch. Rational Mech. Anal., 157, 255-283 (2001) · Zbl 0979.49003
[12] Barron, E. N.; Jensen, R. R.; Wang, C. Y., Lower semicontinuity of L^∞ functionals, Ann. Inst. Henri Poincaré C, 18, 495-517 (2001) · Zbl 1034.49008
[13] Bessail, H.; Olson, E.; Titi, E. S., Continuous data assimilation with stochastically noisy data, Nonlinearity, 28, 729-753 (2015) · Zbl 1308.35161
[14] Bocea, M.; Nesi, V., Γ-convergence of power-law functionals, variational principles in L^∞, and applications, SIAM J. Math. Anal., 39, 1550-1576 (2008) · Zbl 1166.35300
[15] Bocea, M.; Popovici, C., Variational principles in L^∞ with applications to antiplane shear and plane stress plasticity, J. Convex Anal., 18, 403-416 (2011) · Zbl 1223.35040
[16] Bröcker, J., What is the correct cost functional for variational data assimilation?, Clim. Dyn., 52, 389-399 (2019)
[17] Bröecker, J., On variational data assimilation in continuous time, Q. J. R. Meteorol. Soc., 136, 1906-1919 (2010)
[18] Bröecker, J., Existence and uniqueness for variational data assimilation in continuous time (2018)
[19] Bröcker, J.; Kuna, T.; Oljaca, L., Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise, SIAM J. Appl. Dyn. Syst., 17, 2882-2914 (2018) · Zbl 1409.62183
[20] Champion, T.; De Pascale, L.; Prinari, F., Γ-convergence and absolute minimizers for supremal functionals, ESAIM Control Optim. Calc. Var., 10, 14-27 (2004) · Zbl 1068.49007
[21] Chen, G.; Huang, X.; Yang, X., Vector Optimization: Set-Valued and Variational Analysis (2005), Berlin: Springer, Berlin
[22] Crandall, M. G., A visit with the ∞-Laplacian, Calculus of Variations and Non-Linear Partial Differential Equations (2005), Berlin: Springer, Berlin
[23] Croce, G.; Katzourakis, N.; Pisante, G., -solutions to the system of vectorial calculus of variations in L^∞ via the singular value problem, Discrete Contin. Dyn. Syst., 37, 6165-6181 (2017) · Zbl 1386.35081
[24] Dacorogna, B., Direct Methods in the Calculus of Variations (2008), Berlin: Springer, Berlin · Zbl 1140.49001
[25] Di Nezza, E.; Palatucci, G.; Valdinoci, E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136, 521-573 (2012) · Zbl 1252.46023
[26] D’Elia, M.; Perego, M.; Veneziani, A., A variational data assimilation procedure for the incompressible Navier-Stokes equations in hemodynamics, J. Sci. Comput., 52, 340-359 (2012) · Zbl 1264.76076
[27] Florescu, L. C.; Godet-Thobie, C., Young Measures and Compactness in Metric Spaces (2012), Berlin: de Gruyter & Co, Berlin · Zbl 1259.28002
[28] 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, 1-23 (2016) · Zbl 1334.35202
[29] Foias, C.; Mondaini, C. F.; Titi, E. S., A discrete data assimilation scheme for the solutions of the two-dimensional Navier-Stokes equations and their statistics, SIAM J. Appl. Dyn. Syst., 15, 2109-2142 (2016) · Zbl 1362.35208
[30] Garroni, A.; Nesi, V.; Ponsiglione, M., Dielectric breakdown: optimal bounds, Proc. R. Soc. A, 457, 2317-2335 (2001) · Zbl 0993.78015
[31] Gerhardt, C., L^p estimates for solutions to the instationary Navier-Stokes equations in dimension two, Pac. J. Math., 79, 375-398 (1978) · Zbl 0408.76011
[32] Giaquinta, M.; Martinazzi, L., An Introduction to the Regularity Theory for Elliptic Systems, Harmonic Maps and Minimal Graphs (2012), Berlin: Springer, Berlin · Zbl 1262.35001
[33] Giga, Y., Solutions of semilinear parabolic equations in L^p and regularity of weak solutions of the Navier-Stokes equations, J. Diff. Equations, 61, 186-212 (1982) · Zbl 0577.35058
[34] Giga, Y.; Sohr, H., Abstract L^p estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102, 72-94 (1991) · Zbl 0739.35067
[35] Katzourakis, N., An Introduction to Viscosity Solutions for Fully Nonlinear PDE with Applications to Calculus of Variations inL^∞ (2015) · Zbl 1326.35006
[36] Katzourakis, N., Generalised solutions for fully nonlinear PDE systems and existence-uniqueness theorems, J. Differ. Equ., 263, 641-686 (2017) · Zbl 1362.35105
[37] Katzourakis, N., An L^∞ regularisation strategy to the inverse source identification problem for elliptic equations, SIAM J. Math. Anal., 51, 1349-1370 (2019) · Zbl 1414.35066
[38] Katzourakis, N., A minimisation problem in L^∞ with PDE and unilateral constraints, ESAIM Control Optim. Calc. Var., 26, 60 (2020) · Zbl 1451.35240
[39] Katzourakis, N., Inverse optical tomography through PDE-constrained optimisation in L^∞, SIAM J. Control Optim., 57, 4205-4233 (2019) · Zbl 1430.49037
[40] Katzourakis, N.; Moser, R., Existence, uniqueness and structure of second order absolute minimisers, Arch. Ration. Mech. Anal., 231, 1615-1634 (2018) · Zbl 1407.49002
[42] Katzourakis, N.; Pryer, T., On the numerical approximation of p-Biharmonic and ∞-Biharmonic functions, Numer. Methods Partial Differ. Equ., 35, 155-180 (2018) · Zbl 1419.65113
[43] Korn, P., Data assimilation for the Navier-Stokes-equations, Physica D, 238, 1957-1974 (2009) · Zbl 1172.76015
[44] Kreisbeck, C.; Zappale, E., Lower semicontinuity and relaxation of nonlocal L^∞-functionals, Calc. Var. Partial Differ. Equ., 59, 1-36 (2020) · Zbl 1446.49011
[45] Larios, A.; Pei, Y., Approximate continuous data assimilation of the 2D Navier-Stokes equations via the Voigt-regularization with observable data, Evol. Equ. Control Theor., 9, 733-751 (2019) · Zbl 1452.35133
[46] Moser, R.; Schwetlick, H., Minimizers of a weighted maximum of the Gauss curvature, Ann. Global Anal. Geom., 41, 199-207 (2012) · Zbl 1236.53035
[47] Miao, Q.; Wang, C.; Zhou, Y., Uniqueness of absolute minimizers for L^∞-functionals involving Hamiltonians H(x, p), Arch. Ration. Mech. Anal., 223, 141-198 (2017) · Zbl 1356.35013
[48] Prinari, F.; Zappale, E., A relaxation result in the vectorial setting and power law approximation for supremal functionals, J. Optim. Theory Appl., 186, 412-452 (2020) · Zbl 1447.49024
[49] Ribeiro, A. N.; Zappale, E., Existence of minimisers for nonlevel convex functionals, SIAM J. Control Opt., 52, 3341-3370 (2014) · Zbl 1307.49022
[50] Schwarz, A.; Dwight, R. P., Data assimilation for Navier-Stokes using the least-squares finite-element method, Int. J. Uncertain. Quantification, 8, 383-403 (2018)
[51] Solonnikov, V. A., Estimates for solution of nonstationary Navier-Stokes equations, J. Sov. Math., 8, 467-529 (1977) · Zbl 0404.35081
[52] Sohr, H., The Navier-Stokes Equations An Elementary Functional Analytic Approach (2001), Berlin: Springer, Berlin · Zbl 0983.35004
[53] Triebel, H., Interpolation Theory, Function Spaces, Differential Operators (1978), Amsterdam: North-Holland, Amsterdam · Zbl 0387.46032
[54] Triebel, H., Function spaces in Lipschitz domains and on Lipschitz manifolds, characteristic functions as pointwise multipliers, Rev. Mat. Complut., 15, 475-524 (2002) · Zbl 1034.46033
[55] Zeidler, E., Nonlinear Functional Analysis and its Application III: Variational Methods and Optimization (1985), Berlin: Springer, Berlin
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