arXiv subjects

Dallas Albritton

Publications and source records attributed to Dallas Albritton.

25 records · Page 2Linked to original sources

arXiv2021

Regularity properties of passive scalars with rough divergence-free drifts

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t θ- Δθ+ b \cdot \nabla θ= 0 \] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $\frac{2}{q} + \frac{n}{p} < 2$, or $L^p_x L^q_t$, where $\frac{3}{q} + \frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of drifts with `bounded total speed' is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before' they can be dissipated.

math.AP↗

arXiv2021

Enhanced dissipation and Hörmander's hypoellipticity

We examine the phenomenon of enhanced dissipation from the perspective of Hörmander's classical theory of second order hypoelliptic operators [31]. Consider a passive scalar in a shear flow, whose evolution is described by the advection-diffusion equation \[ \partial_t f + b(y) \partial_x f - νΔf = 0 \text{ on } \mathbb{T} \times (0,1) \times \mathbb{R}_+ \] with periodic, Dirichlet, or Neumann conditions in $y$. We demonstrate that decay is enhanced on the timescale $T \sim ν^{-(N+1)/(N+3)}$, where $N-1$ is the maximal order of vanishing of the derivative $b'(y)$ of the shear profile and $N=0$ for monotone shear flows. In the periodic setting, we recover the known timescale of Bedrossian and Coti Zelati [8]. Our results are new in the presence of boundaries.

math.AP↗

arXiv2021

Long-time behavior of scalar conservation laws with critical dissipation

The critical Burgers equation $\partial_t u + u \partial_x u + Λu = 0$ is a toy model for the competition between transport and diffusion with regard to shock formation in fluids. It is well known that smooth initial data does not generate shocks in finite time. Less is known about the long-time behavior for `shock-like' initial data: $u_0 \to \pm a$ as $x \to \mp \infty$. We describe this long-time behavior in the general setting of multidimensional critical scalar conservation laws $\partial_t u + \text{div}f(u) + Λu = 0$ when the initial data has limits at infinity. The asymptotics are given by certain self-similar solutions, whose stability we demonstrate with the optimal diffusive rates.

math.AP↗

arXiv2019

Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

We generalize two results in the Navier-Stokes regularity theory whose proofs rely on `zooming in' on a presumed singularity to the local setting near a curved portion $Γ\subset \partialΩ$ of the boundary. Suppose that $u$ is a boundary suitable weak solution with singularity $z^* = (x^*,T^*)$, where $x^* \in Ω\cup Γ$. Then, under weak background assumptions, the $L_3$ norm of $u$ tends to infinity in every ball centered at $x^*$: \begin{equation*} \lim_{t \to T^*_-} \lVert u(\cdot, t)\rVert_{L_{3}\left(Ω\cap B(x^*,r)\right)} = \infty \quad \forall r > 0. \end{equation*} Additionally, $u$ generates a non-trivial `mild bounded ancient solution' in $\mathbb{R}^3$ or $\mathbb{R}^3_+$ through a rescaling procedure that `zooms in' on the singularity. Our proofs rely on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for $L_3$ initial data and a Liouville theorem. For the latter result, we apply perturbation theory for $L_\infty$ initial data based on linear estimates due to K. Abe and Y. Giga.

math.AP↗

arXiv2019

On local Type I singularities of the Navier-Stokes equations and Liouville theorems

We prove that suitable weak solutions of the Navier-Stokes equations exhibit Type I singularities if and only if there exists a non-trivial mild bounded ancient solution satisfying a Type I decay condition. The main novelty is in the reverse direction, which is based on the idea of zooming out on a regular solution to generate a singularity. By similar methods, we prove a Liouville theorem for ancient solutions of the Navier-Stokes equations bounded in $L^3$ along a backward sequence of times.

math.AP↗

arXiv2018

Global weak Besov solutions of the Navier-Stokes equations and applications

We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy a certain stability property with respect to the weak-$\ast$ convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.

math.AP↗

arXiv2018

Blow-up criteria for the Navier-Stokes equations in non-endpoint critical Besov spaces

We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T^* < \infty$, then the non-endpoint critical Besov norms must become infinite at the blow-up time: $$ \lim_{t \uparrow T^*} \lVert{u(\cdot,t)}\rVert_{\dot B^{-1+3/p}_{p,q}(\mathbb{R}^3)} = \infty, \quad 3 < p,q < \infty. $$ In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

math.AP↗