Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userStrong well-posedness, stability and optimal control theory for a mathematical model for magneto-viscoelastic fluids
142
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this article, we study the strong well-posedness, stability and optimal control of an incompressible magneto-viscoelastic fluid model in two dimensions. The model consists of an incompressible Navier--Stokes equation for the velocity field, an evolution equation for the deformation tensor, and a gradient flow equation for the magnetization vector. First, we prove that the model under consideration posseses a global strong solution in a suitable functional framework. Second, we derive stability estimates with respect to an external magnetic field. Based on the stability estimates we use the external magnetic field as the control to minimize a cost functional of tracking-type. We prove existence of an optimal control and derive first-order necessary optimality conditions. Finally, we consider a second optimal control problem, where the external magnetic field, which represents the control, is generated by a finite number of fixed magnetic field coils.
rate research
Read More
We provide a new analysis of the Boltzmann equation with constant collision kernel in two space dimensions. The scaling-critical Lebesgue space is $L^2_{x,v}$; we prove global well-posedness and a version of scattering, assuming that the data $f_0$ is sufficiently smooth and localized, and the $L^2_{x,v}$ norm of $f_0$ is sufficiently small. The proof relies upon a new scaling-critical bilinear spacetime estimate for the collision gain term in Boltzmanns equation, combined with a novel application of the Kaniel-Shinbrot iteration.
We investigate a mathematical theory for the erosion of sediment which begins with the study of a non-linear, parabolic, weighted 4-Laplace equation on a rectangular domain corresponding to a base segment of an extended landscape. Imposing natural boundary conditions, we show that the equation admits entropy solutions and prove regularity and uniqueness of weak solutions when they exist. We then investigate a particular class of weak solutions studied in previous work of the first author and produce numerical simulations of these solutions. After introducing an optimal transportation problem for the sediment flow, we show that this class of weak solutions implements the optimal transportation of the sediment.
Considering the Cauchy problem for the modified finite-depth-fluid equation $partial_tu-G_delta(partial_x^2u)mp u^2u_x=0, u(0)=u_0$, where $G_delta f=-i ft ^{-1}[coth(2pi delta xi)-frac{1}{2pi delta xi}]ft f$, $deltages 1$, and $u$ is a real-valued function, we show that it is uniformly globally well-posed if $u_0 in H^s (sgeq 1/2)$ with $ orm{u_0}_{L^2}$ sufficiently small for all $delta ges 1$. Our result is sharp in the sense that the solution map fails to be $C^3$ in $H^s (s<1/2)$. Moreover, we prove that for any $T>0$, its solution converges in $C([0,T]; H^s)$ to that of the modified Benjamin-Ono equation if $delta$ tends to $+infty$.
In this paper we show global well-posedness near vacuum for the binary-ternary Boltzmann equation. The binary-ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel-Shinbrot iteration and related work to approximate the solution of the nonlinear equation by monotone sequences of supersolutions and subsolutions. This analysis required establishing new convolution type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.
We prove that the complex-valued modified Benjamin-Ono (mBO) equation is locally wellposed if the initial data $phi$ belongs to $H^s$ for $sgeq 1/2$ with $ orm{phi}_{L^2}$ sufficiently small without performing a gauge transformation. Hence the real-valued mBO equation is globally wellposed for those initial datas, which is contained in the results of C. Kenig and H. Takaoka cite{KenigT} where the smallness condition is not needed. We also prove that the real-valued $H^infty$ solutions to mBO equation satisfy a priori local in time $H^s$ bounds in terms of the $H^s$ size of the initial data for $s>1/4$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
