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The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli's law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.
The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli's law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.
We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings W 0 s , p ( Ω ) ↪ L q ( Ω ) , {W}_{0}^{s,p}(\Omega )\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{q}(\Omega ), where N ≥ 1 N\ge 1 , 0 < s < 1 0\lt s\lt 1 , p = 1 , 2 p=1,2 , 1 ≤ q < p s ∗ = N p N − s p 1\le q\lt {p}_{s}^{\ast }=\frac{Np}{N-sp} , and Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded smooth domain or the whole space R N {{\mathbb{R}}}^{N} . Our results cover the borderline case p = 1 p=1 , the Hilbert case p = 2 p=2 , N > 2 s N\gt 2s , and the so-called Sobolev limiting case N = 1 N=1 , s = 1 2 s=\frac{1}{2} , and p = 2 p=2 , where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.
<abstract><p>We give a new full explanation of the Tacoma Narrows Bridge collapse, occurred on November 7, 1940. Our explanation involves both structural phenomena, such as parametric resonances, and sophisticated mathematical tools, such as the Floquet theory. Contrary to all previous attempts, our explanation perfectly fits, both qualitatively and quantitatively, with what was observed that day. With this explanation at hand, we set up and partially solve some optimal control and shape optimization problems (both analytically and numerically) aiming to improve the stability of bridges. The control parameter to be optimized is the strength of a partial damping term whose role is to decrease the energy within the deck. Shape optimization intends to give suggestions for the design of future bridges.</p></abstract>
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