arXiv subjects
Publications and source records attributed to Benjamin Ingimarson.
We consider the Kuramoto-Sivashinsky equation $u_t+uu_x+u_{xx}+u_{xxxx}=0$ with $L$-periodic boundary conditions and zero spatial mean. We prove that every smooth solution satisfies $\|u(t)\|_{L^2}\leq CL^{5/6}+CL^{3/2}/t$ for $t>0$, where $C$ is a universal constant. In particular, $\|u(t)\|_{L^2}\leq CL^{5/6}$ for all $t\geq L^{2/3}$, independently of the initial norm.
We answer a question of Constantin, Ignatova, and Vicol concerning regularity of asymptotically axisymmetric solutions to the three-dimensional Navier-Stokes equations. For suitable weak solutions that are smooth before a terminal time and have a force uniformly bounded in $C_x^2$, we replace spatial analyticity of the force by rotational invariance of its curl. Under the same anisotropic Type II bounds on the angular mean and exact axial symmetry on a possibly collapsing core, the solution is axisymmetric on the entire cylinder and regular at the putative singular point. No time regularity or quantitative lower bound on the core radius is required. The proof combines unique continuation for linear Stokes systems with a Baire category argument and the axisymmetric regularity theorem due to Constantin, Ignatova, and Vicol.
We prove the persistence of conormal analyticity through the $H^r$ norm, with $r \geq 4$, for the 3D incompressible Euler equations in the half-space $Ω= \mathbb{R}_+^3$. We also establish a lower bound on the radius of conormal analyticity which depends on $\|u\|_{H^r}$ and initial data. In particular, persistence is established in both of the spaces $X_τ^{\rm co}$ and $A_τ^{\rm co}$, the former being based on the conormal operator $Z_k = ϕ^k \partial_3^k$ and the latter being based on $L^k = (ϕ\partial_3)^k$. Finally, we show that the norms $\|\cdot \|_{X_τ^{\rm co}}$ and $\|\cdot \|_{A_τ^{\rm co}}$ are equivalent up to a small loss in radius, the proof of which uses explicit identities relating the two conormal operators.
Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^kω\|_{L^\infty(Ω)}$ for $k \geq 1$. We also show that the Gronwall-type inequality satisfied by $\|ω(t)\|_{L^\infty}$, in the cases that $Ω= \mathbb{R}^3$, $\mathbb{T}^3$, or a bounded domain, exhibits wildly oscillating solutions.
Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In particular, when the domain is $\mathbb{R}^3$ or $\mathbb{T}^3$, we provide lower bounds on $\int_{0}^{t}\Vert ω\Vert_{L^\infty}\,ds$ and $\sup_{s\in[0,t]}\|ω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide pointwise-in-time lower bounds on~$\|D^k ω\|_{L^\infty}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor.
We establish local-in-time existence for the Euler equations on a bounded domain with space-time dependent variable coefficients, given initial data $v_0 \in H^r$ under the optimal regularity condition $r > 2.5$. In the case $r = 3$, we further prove a Beale-Kato-Majda criterion that relates blow-up in the $H^r$ norm to the BMO norm of the variable vorticity $ζ$.
We prove the existence of small solitary waves for one-dimensional lattices of particles that each repel every other particle with a force that decays as a power of distance. For force exponents $α+1$ with $\frac43<α<3$, we employ fixed-point arguments to find near-sonic solitary waves having scaled velocity profiles close to non-degenerate solitary-wave profiles of fractional KdV or generalized Benjamin-Ono equations. These equations were recently found to approximately govern unidirectional long-wave motions in these lattices.
We study waves on infinite one-dimensional lattices of particles that each interact with all others through power-law forces $F \sim r^{-β}$. The inverse-cube case corresponds to Calogero-Moser systems, which are well known to be completely integrable for any finite number of particles. The formal long-wave limit for unidirectional waves in these lattices is the Korteweg-de Vries equation if $β>4$, but with $2<β<4$ it is a nonlocal dispersive PDE that reduces to the Benjamin-Ono equation for $β=3$. For the infinite Calogero-Moser lattice, we find explicit formulas that describe solitary and periodic traveling waves.
We study convergence of the spectral shift function for the finite interval restrictions of a pair of full-line Schrödinger operators to an interval of the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as $\ell\to \infty$ in the case when the finite interval restrictions are relatively prime to those with Dirichlet boundary conditions. Using a Krein-type resolvent identity we show that the spectral shift function for the finite interval restrictions converges weakly to that for the pair of full-line Schrödinger operators as the length of the interval tends to infinity.