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Global existence and maximal regularity of solutions of gradient systems. (English) Zbl 1209.47020
The author applies the Galerkin method in order to prove a maximal regularity result for the following abstract gradient system
\[ \begin{cases} u'(t)+\nabla_g E(u(t))=f(t)\quad \text{for a.a.}\;t\in[0,T],\\ u(0)=u_0. \end{cases} \]
Here, \(E: V\to\mathbb R\) is a continuously differentiable function over a real Banach space \(V\) densely and compactly embedded into a real Hilbert space \(H,\) \(\nabla_g E\) stands for the gradient of \(E\) with respect to some metric \(g,\) \(f\in L^2(0,T;H)\) and \(u_0\in V.\)
The abstract result is applied to nonlinear diffusion equations and to nondegenerate quasilinear parabolic equations with nonlocal coefficients.

MSC:
47E05 General theory of ordinary differential operators
35Q99 Partial differential equations of mathematical physics and other areas of application
35B65 Smoothness and regularity of solutions to PDEs
35G10 Initial value problems for linear higher-order PDEs
47A60 Functional calculus for linear operators
47J35 Nonlinear evolution equations
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[1] W. Arendt, R. Chill, Global existence for quasilinear diffusion equations in isotropic nondivergence form, Ann. Sc. Norm. Super. Pisa Cl. Sci., in press. · Zbl 1223.35202
[2] Amann, H., Quasilinear evolution equations and parabolic systems, Trans. amer. math. soc., 293, 1, 191-227, (1986) · Zbl 0635.47056
[3] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient flows in metric spaces and in the space of probability measures, (2003), Birkhäuser Verlag Basel, Boston, Berlin
[4] Arendt, W., Semigroups and evolution equations: functional calculus, regularity and kernel estimates, (), 1-85 · Zbl 1082.35001
[5] Brézis, H., Analyse fonctionnelle. théorie et applications, (1999), Dunod Paris
[6] Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, North-holland math. stud., vol. 5, (1973), North-Holland Amsterdam, London · Zbl 0252.47055
[7] Evans, L.C., Partial differential equations, Grad. stud. math., vol. 19, (2010), American Mathematical Society Providence, RI
[8] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill Book Company New York, Toronto, London · Zbl 0042.32602
[9] Denk, R.; Hieber, M.; Prüss, J., \(\mathcal{R}\)-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. amer. math. soc., 166, 788, (2003), viii+114 pp
[10] Hille, E., Functional analysis and semi-groups, Colloquium publication, vol. 31, (1948), American Mathematical Society · Zbl 0033.06501
[11] Kunstmann, P.C.; Weis, L., Maximal \(L^p\)-regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty\)-functional calculus, (), 65-311 · Zbl 1097.47041
[12] Ladyženskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear equations of parabolic type, Transl. math. monogr., vol. 23, (1967), American Mathematical Society Providence, RI
[13] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (2002), Dunod Paris · Zbl 0189.40603
[14] Lunardi, A., Analytic semigroups and optimal regularity in parabolic problems, Progr. nonlinear differential equations appl., vol. 16, (1995), Birkhäuser Basel · Zbl 0816.35001
[15] Otto, F., The geometry of dissipative evolution equations: the porous medium equation, Comm. partial differential equations, 26, 1-2, 101-174, (2001) · Zbl 0984.35089
[16] Ovono, A.A.; Rougirel, A., Elliptic equations with diffusion parametrized by the range of nonlocal interactions, Ann. mat. pura appl., 189, 1, 163-183, (2010) · Zbl 1181.35103
[17] Showalter, R.E., Monotone operators in Banach spaces and nonlinear partial differential equations, Math. surveys monogr., vol. 49, (1997), American Mathematical Society Providence, RI · Zbl 0870.35004
[18] Simon, J., Compact sets in the space \(L^p(0, T; B)\), Ann. mat. pura appl. (4), 146, 65-96, (1987) · Zbl 0629.46031
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