In this short section we address the question of the wellposedness of system (RFε) and we prove Theorem 2.1. This system is very like the three dimensional Navier–Stokes system, for which it is well known that global (possibly not unique) weak solutions exist if the initial data is of...
Solving Stokes' Theorem: Find \int_{\partial S} F \cdot ds Here is the problem: S is the ellipsoid x^2+y^2+2z^2=10 and F is a vector field F=(sin(xy),e^x,-yz) Find: \int \int_S ( \nabla \mbox {x} F) \cdot dS So, I know that Stokes' Theorem states that: \int...
The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit ...
To directly determine the total scattering cross section, the scattered flux in a closed surface surrounding the oligomer was calculated using the Poynting theorem. The surface charge density distributions are obtained by considering the skin effect and applying Gauss’ law during FEM calculations92. ...
In this section, we prove Theorem 2.12. We first show the global well-posedness of (1.1). To this end, we assume n−1<p<∞. By virtue of this restriction, it follows that −1+1/p<−1+n/p<1/p. Step 1: Global existence Define Xp as well as BXp(R1) byXp≡{u∈BC([0...
From [34, Theorem 4.2], we thus conclude that $$\begin{aligned} W_\infty (V_0^{1+\varepsilon },(V_0^{1-\varepsilon })')\hookrightarrow C_b([0,\infty );V_0^\varepsilon ), \end{aligned}$$ where \(C_b\) denotes the space of continuous and bounded functions. ...
where the Kelvin circulation theorem is valid3,17. Thus, it is evident that a non-potential or rotational flow bears vorticity, but to produce vortices in Stokes flow requires substantial efforts due to a strong dissipation of vorticity at{\mathrm{Re}} \ll 1. Strikingly, in a wall-dominated...
To be fair, there is some work applying the fluctuation-dissipation theorem to hydrodynamics- a good summary is in Chapter 8 of Chaikin and Lubensky's "Principles of condensed matter physics". Dattagupta and Puri "Dissipative phenomena in condensed matter" also has a good summary of current res...
Theorem . (see [, , ]) Let X, Y be two Banach spaces. X∗, Y ∗ be, respectively, their dual spaces. Assume that X is dense in Y , the injection i : X → Y is continuous, its adjoint i∗ : Y ∗ → X∗ is dense, and U is a norm...
In this section, we present the proof of Theorem 1.1. Instead of proving the existence of the solution to (1.6) directly, we consider an alternative ODE system (3.1) to which Proposition 2.1 can be applied. We will first show the existence and uniqueness of the solution to the ODE system...