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Register a new userLower bounds on mixing norms for the advection diffusion equation in $mathbb{R}^d$
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An algebraic lower bound on the energy decay for solutions of the advection-diffusion equation in $mathbb{R}^d$ with $d=2,3$ is derived using the Fourier splitting method. Motivated by a conjecture on mixing of passive scalars in fluids, a lower bound on the $L^2-$ norm of the inverse gradient of the solution is obtained via gradient estimates and interpolation.
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We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term $u^3$ and potential periodic in $t$ may have solutions with exponentially increasing as $ t to infty$ norm $H^1({mathbb R}^3_x)$. In [2] it was established that adding the nonlinear term $u^3$ the $H^1({mathbb R}^3_x)$ norm of the solution is polynomially bounded for every choice of $q$. In this paper we show that $H^k({mathbb R}^3_x)$ norm of this global solution is also polynomially bounded. To prove this we apply a different argument based on the analysis of a sequence ${Y_k(ntau_k)}_{n = 0}^{infty}$ with suitably defined energy norm $Y_k(t)$ and $0 < tau_k <1.$
We prove global well-posedness for the $3D$ radial defocusing cubic wave equation with data in $H^{s} times H^{s-1}$, $1>s>{7/10}$.
We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.
In this paper, we use Dafermos-Rodnianskis new vector field method to study the asymptotic pointwise decay properties for solutions of energy subcritical defocusing semilinear wave equations in $mathbb{R}^{1+3}$. We prove that the solution decays as quickly as linear waves for $p>frac{1+sqrt{17}}{2}$, covering part of the sub-conformal case, while for the range $2<pleq frac{1+sqrt{17}}{2}$, the solution still decays with rate at least $t^{-frac{1}{3}}$. As a consequence, the solution scatters in energy space when $p>2.3542$. We also show that the solution is uniformly bounded when $p>frac{3}{2}$.
We study mass preserving transport of passive tracers in the low-diffusivity limit using Lagrangian coordinates. Over finite-time intervals, the solution-operator of the nonautonomous diffusion equation is approximated by that of a time-averaged diffusion equation. We show that leading order asymptotics that hold for functions [Krol, 1991] extend to the dominant nontrivial singular value. This answers questions raised in [Karrasch & Keller, 2020]. The generator of the time-averaged diffusion/heat semigroup is a Laplace operator associated to a weighted manifold structure on the material manifold. We show how geometrical properties of this weighted manifold directly lead to physical transport quantities of the nonautonomous equation in the low-diffusivity limit.
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