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Necessary and sufficient conditions for the fractional Gagliardo-Nirenberg inequalities and applications to Navier-Stokes and generalized Boson equations. (English) Zbl 1270.42026
The Gagliardo-Nirenberg (GN) inequality is a fundamental tool in the study of nonlinear partial differential equations. In this paper, the authors obtain necessary and sufficient conditions for the GN inequality in homogeneous Besov spaces and Triebel-Lizorkin spaces. As the first application, the authors show that the solution of the Navier-Stokes equations at finite blowup time \(T_m\) has concentration phenomena in the critical space \(L^3({\mathbb{R}}^3)\). As the second application, the authors consider the minimization problem for the variational problem \[ M_c = \inf \{ E(u): \|u_i\|_2^2 = c_i > 0, i = 1, \dots, L\} \] where \[ E(u) = \frac{1}{2} \|u \|_{{\dot{H}}_s}^2 - \int_{{\mathbb{R}}^{2n}} G(u(x)) V(x - y) G(u(y)) dx dy \] for \(u = (u_1, \dots, u_{L})\). The authors show that \(M_c\) admits a radial and radially decreasing minimizer under suitable assumptions an \(s, G\) and \(V\).

MSC:
42B35 Function spaces arising in harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35J50 Variational methods for elliptic systems
35Q40 PDEs in connection with quantum mechanics
47J30 Variational methods involving nonlinear operators
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