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Register a new userRenormalization and energy conservation for axisymmetric fluid flows
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We study vanishing viscosity solutions to the axisymmetric Euler equations with (relative) vorticity in $L^p$ with $p>1$. We show that these solutions satisfy the corresponding vorticity equations in the sense of renormalized solutions. Moreover, we show that the kinetic energy is preserved provided that $p>3/2$ and the vorticity is nonnegative and has finite second moments.
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We consider the compressible Navier--Stokes equation in a perturbed half-space with an outflow boundary condition as well as the supersonic condition. For a half-space, it has been known that a certain planar stationary solution exist and it is time-asymptotically stable. The planar stationary solution is independent of the tangential directions and its velocities of the tangential directions are zero. In this paper, we show the unique existence of stationary solutions for the perturbed half-space. The feature of our work is that our stationary solution depends on all directions and has multidirectional flow. Furthermore, we also prove the asymptotic stability of this stationary solution.
This paper addresses the mathematical models for the heat-conduction equations and the Navier-Stokes equations via fractional derivatives without singular kernel.
We study linear inhomogeneous kinetic equations with an external confining potential and a collision operator with several local conservation laws (local density, momentum and energy). We exhibit all equilibria and entropy-maximizing special modes, and we prove asymptotic exponential convergence of solutions to them with quantitative rate. This is the first complete picture of hypocoercivity and quantitative $H$-theorem for inhomogeneous kinetic equations in this setting.
Generalizing results by Bryant and Griffiths [Duke Math. J., 1995, V.78, 531-676], we completely describe local conservation laws of second-order (1+1)-dimensional evolution equations up to contact equivalence. The possible dimensions of spaces of conservation laws prove to be 0, 1, 2 and infinity. The canonical forms of equations with respect to contact equivalence are found for all nonzero dimensions of spaces of conservation laws.
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 $+infty$.
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