The regularity of the boundary of a multidimensional aggregation patch

Let $d \geq 2$ and let $N(y)$ be the fundamental solution of the Laplace equation in $R^d$ We consider the aggregation equation $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho v) =0, v = -\nabla N * \rho$$ with initial data $\rho(x,0) = \chi_{D_0}$, where $\chi_{D_0}$ is the indicator function of a bounded domain $D_0 \subset R^d.$ We now fix $0 < \gamma < 1$ and take $D_0$ to be a bounded $C^{1+\gamma}$ domain (a domain with smooth boundary of class $C^{1+\gamma}$). Then we have Theorem: If $D_0$ is a $C^{1 + \gamma}$ domain, then the initial value problem above has a solution given by $$\rho(x,t) = \frac{1}{1 -t} \chi_{D_t}(x), \quad x \in R^d, \quad 0 \le t < 1$$ where $D_t$ is a $C^{1 + \gamma}$ domain for all $0 \leq t < 1$

VISUALIZING BARRIER DUNE TOPOGRAPHIC STATE SPACE AND INFERENCE OF RESILIENCE PROPERTIES

The linkage between barrier island morphologies and dune topographies, vegetation, and biogeomorphic feedbacks, has been examined. The two-fold stability domain (i.e., overwash-resisting and overwash-reinforcing stability domains) model from case studies in a couple of islands along the Georgia Bight and Virginia coast has been proposed to examine the resilience properties in the barrier dune systems. Thus, there is a need to examine geographic variations in the dune topography among and within islands. Meanwhile, previous studies just analyzed and compared dune topographies based on transect-based point elevations or dune crest elevations; therefore, it is necessary to further examine dune topography in terms of multiple patterns and processes across scales. In this dissertation, I develop and deploy a cross-scale data model developed from resilience theory to represent and compare dune topographies across twelve islands over approximately 2,050 kilometers of the US southeastern Atlantic coast. Three sets of topographic variables were employed to summarize the cross-scale structure of topography (elevational statistics, patch indices, and the continuous surface properties). These metrics differed in their degree of spatial explicitness, their level of measurement, and association with patch or gradient paradigms. Topographic metrics were derived from digital elevation models (DEMs) of dune topographies constructed from airborne Light Detection and Ranging (LiDAR). These topographic metrics were used to construct dune topographic state space to investigate and visualize the cross-scale structure of dune topography. This study investigated (1) dune topography and landscape similarity among barrier islands in different barrier island morphologic contexts, (2) the differences in barrier island dune topographies and their resilience properties across large geographic extents, and (3) how geomorphic and biogeomorphic processes are related to resilience prosperities. The findings are summarized below. First, dune topography varies according to island morphologies of the Virginia coast; however, local controls (such as human modification of the shore or shoreline accretion and erosion) also play an important role in shaping dune topographies. Compared with tide-dominated islands, wave-dominated islands exhibited more convergence in dune topographies. Second, the dune landscapes of the Virginia Barrier Islands have a poorly consistent spatial structure, along with strong collinearity among elevational variables and landscape indices, which reflects the rapid retreat and erosion along the coast. The dune landscapes of the Georgia Bight have a more consistent spatial structure and a greater dimensionality in state space. Thus, the weaker multicollinearity and higher dimensionality in the dataset reflect their potential for resilience. Last, islands of different elevations may have similar dune topography characteristics due to the difference in resistance and resilience. Notwithstanding the geographic variability in geomorphic and biogeomorphic processes, convergence in dune topography exists, which is evidenced by the response curves of the topographic metrics that are correlated with both axes. This work demonstrates the usefulness of different representations of dune topography by cross-scale data modeling. Also, the two existing models of barrier island dune states were integrated to form a conceptual model that illuminates different, but complementary, resilience properties in the barrier dune system. The differences in dune topographies and resilience properties were detected in state space, and this information offers guidance for future study’s field site selections

Equilibria of biological aggregations with nonlocal repulsive-attractive interactions

We consider the aggregation equation $\rho_{t}-\nabla\cdot(\rho\nabla K\ast\rho) =0$ in $\mathbb{R}^{n}$, where the interaction potential $K$ incorporates short-range Newtonian repulsion and long-range power-law attraction. We study the global well-posedness of solutions and investigate analytically and numerically the equilibrium solutions. We show that there exist unique equilibria supported on a ball of $\mathbb{R}^n$. By using the method of moving planes we prove that such equilibria are radially symmetric and monotone in the radial coordinate. We perform asymptotic studies for the limiting cases when the exponent of the power-law attraction approaches infinity and a Newtonian singularity, respectively. Numerical simulations suggest that equilibria studied here are global attractors for the dynamics of the aggregation model

Characterization of radially symmetric finite time blowup in multidimensional aggregation equations,

This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range This paper studies the transport of a mass $\mu$ in $\real^d, d \geq 2,$ by a flow field $v= -\nabla K*\mu$. We focus on kernels $K=|x|^\alpha/ \alpha$ for $2-d\leq \alpha<2$ for which the smooth densities are known to develop singularities in finite time. For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time. The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities. We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\alpha >2$ where radially symmetric solutions are known to lose monotonicity. In the case of the Newtonian potential ($\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line. This enables us to prove preservation of monotonicity using the classical theory of conservation laws. In the case $2 -d < \alpha < 2$ and at the critical exponent $p$ we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity. This extends recent work on the local ill-posedness of solutions at the critical exponent.Comment: 30 page