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Register a new userFinite-parameter feedback stabilization of original Burgers equations and Burgers equation with nonlocal nonlinearities
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We study the problem of global exponential stabilization of original Burgers equations and the Burgers equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as $trightarrow infty$ with an exponential rate.
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The formation of singularities in finite time in non-local Burgers equations, with time-fractional derivative, is studied in detail. The occurrence of finite time singularity is proved, revealing the underlying mechanism, and precise estimates on the blow-up time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.
The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{frac{alpha}{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state solution, will return to the steady state as $trightarrowinfty$.
We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $Rtimes (0,infty)$, $u(x,0)=u_0(x)ge 0$ in $R$, as $mtoinfty$ for any $p>1$ and $u_0in L^1(R)cap L^{infty}(R)$ or as $ptoinfty$ for any $m>1$ and $u_0in L^{infty}(R)$ . We also show that in general $underset{ptoinfty}limunderset{mtoinfty}lim u_{m,p} eunderset{mtoinfty}limunderset{ptoinfty}lim u_{m,p}$.
In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in $L^infty(L^2)$, $L^infty(H^1)$ and $L^infty(L^infty)$-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.
In this article, global stabilization results for the two dimensional (2D) viscous Burgers equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $C^0$-conforming finite element method in spatial direction, optimal error estimates in $L^infty(L^2)$ and in $L^infty(H^1)$- norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.
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