Analysis of PDEs (math.AP)

Abstract: We study the long-time behavior of scalar viscous conservation laws via the structure of $\omega$-limit sets. We show that $\omega$-limit sets always contain constants or shocks by establishing convergence to shocks for arbitrary monotone initial data. In the particular case of Burgers' equation, we review and refine results that parametrize entire solutions in terms of probability measures, and we construct initial data for which the $\omega$-limit set is not reduced to the translates of a single shock. Finally we propose several open problems related to the description of long-time dynamics.

Abstract: The morphology of crystalline thin films evolving on flat rigid substrates by condensation of extra film atoms or by evaporation of their own atoms in the surrounding vapor is studied in the framework of the theory of Stress Driven Rearrangement Instabilities (SDRI). By following the SDRI literature both the elastic contributions due to the mismatch between the film and the substrate lattices at their theoretical (free-standing) elastic equilibrium, and a curvature perturbative regularization preventing the problem to be ill-posed due to the otherwise exhibited backward parabolicity, are added in the evolution equation. The resulting Cauchy problem under investigation consists in an anisotropic mean-curvature type flow of the fourth order on the film profiles, which are assumed to be parametrizable as graphs of functions measuring the film thicknesses, coupled with a quasistatic elastic problem in the film bulks. Periodic boundary conditions are considered. The results are twofold: the existence of a regular solution for a finite period of time and the stability for all times, of both Lyapunov and asymptotic type, of any configuration given by a flat film profile and the related elastic equilibrium. Such achievements represent both the generalization to three dimensions of a previous result in two dimensions for a similar Cauchy problem, and the complement of the analysis previously carried out in the literature for the symmetric situation in which the film evolution is not influenced by the evaporation-condensation process here considered, but it is entirely due to the volume preserving surface-diffusion process, which is instead here neglected. The method is based on minimizing movements, which allow to exploit the the gradient-flow structure of the evolution equation.

Abstract: We study the Lane-Emden problem \[\begin{cases} -\Delta u_p =|u_p|^{p-1}u_p&\text{in}\quad \Omega, u_p=0 &\text{on}\quad\partial\Omega, \end{cases}\] where $\Omega\subset\mathbb R^2$ is a smooth bounded domain and $p>1$ is sufficiently large. We obtain sharp estimates and non-degeneracy of low energy nodal solutions $u_p$ (i.e. nodal solutions satisfying $\lim_{p\to+\infty}p\int_{\Omega}|u_p|^{p+1}dx=16\pi e$). As applications, we prove that the comparable condition $p(\|u_p^+\|_{\infty}-\|u_p^-\|_{\infty})=O(1)$ holds automatically for least energy nodal solutions, which confirms a conjecture raised by (Grossi-Grumiau-Pacella, Ann.I.H. Poincare-AN, 30 (2013), 121-140) and (Grossi-Grumiau-Pacella, J.Math.Pures Appl. 101 (2014), 735-754).

Abstract: We consider the Vlasov--Poisson system describing a two-species plasma with spatial dimension $1$ and the velocity variable in $\mathbb{R}^n$. We find the necessary and sufficient conditions for the existence of solitary waves, shock waves, and wave trains of the system, respectively. To this end, we need to investigate the distribution of ions trapped by the electrostatic potential. Furthermore, we classify completely in all possible cases whether or not the traveling wave is unique. The uniqueness varies according to each traveling wave when we exclude the variant caused by translation. For the solitary wave, there are both cases that it is unique and nonunique. The shock wave is always unique. No wave train is unique.

Abstract: The Koopman--von Neumann equation describes the evolution of a complex-valued wavefunction corresponding to the probability distribution given by an associated classical Liouville equation. Typically, it is defined on the whole Euclidean space. The investigation of bounded domains, particularly in practical scenarios involving quantum-based simulations of dynamical systems, has received little attention so far. We consider the Koopman--von Neumann equation associated with an ordinary differential equation on a bounded domain whose trajectories are contained in the set's closure. Our main results are the construction of a strongly continuous semigroup together with the existence and uniqueness of solutions of the associated initial value problem. To this end, a functional-analytic framework connected to Sobolev spaces is proposed and analyzed. Moreover, the connection of the Koopman--von Neumann framework to transport equations is highlighted.

Abstract: This paper fully answers a long standing open question concerning the stability/instability of pure gravity periodic traveling water waves -- called Stokes waves -- at the critical Whitham-Benjamin depth $ \mathtt{h}_{\scriptscriptstyle WB} = 1.363... $ and nearby values. We prove that Stokes waves of small amplitude $ \mathcal{O}( \epsilon ) $ are, at the critical depth $ \mathtt{h}_{\scriptscriptstyle WB} $, linearly unstable under long wave perturbations. This is also true for slightly smaller values of the depth $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} - c \epsilon^2 $, $ c > 0 $, depending on the amplitude of the wave. This problem was not rigorously solved in previous literature because the expansions degenerate at the critical depth. In order to resolve this degenerate case, and describe in a mathematically exhaustive way how the eigenvalues change their stable-to-unstable nature along this shallow-to-deep water transient, we Taylor expand the computations of arXiv:2204.00809v2 at a higher degree of accuracy, derived by the fourth order expansion of the Stokes waves. We prove that also in this transient regime a pair of unstable eigenvalues depict a closed figure "8", of smaller size than for $ \mathtt{h} > \mathtt{h}_{\scriptscriptstyle WB} $, as the Floquet exponent varies.

Abstract: Given $\lambda>0$ and $p>2$, we present a complete classification of the positive $H^1$-solutions of the equation $-u''+\lambda u=|u|^{p-2}u$ on the $\mathcal{T}$-metric graph (consisting of two unbounded edges and a terminal edge of length $\ell>0$, all joined together at a single vertex). This study implies, in particular, the uniqueness of action ground states and that, for $p\sim 6^-$, the notions of action and energy ground states do not coincide in general. In the $L^2$-supercritical case $p>6$, we prove that, for both $\lambda$ small and large, action ground states are orbitally unstable for the flow generated by the associated time-dependent NLS equation $i\partial_tu + \partial^2_{xx} u + |u|^{p-2}u=0$. Finally, we provide numerical evidence of the uniqueness of energy ground states for $p<6$ and of the existence of both stable and unstable action ground states for $p\sim6$.

Abstract: In this paper, we investigate the stability of the 2-dimensional (2D) Taylor-Couette (TC) flow for the incompressible Navier-Stokes equations. The explicit form of velocity for 2D TC flow is given by $u=(Ar+\frac{B}{r})(-\sin \theta, \cos \theta)^T$ with $(r, \theta)\in [1, R]\times \mathbb{S}^1$ being an annulus and $A, B$ being constants. Here, $A, B$ encode the rotational effect and $R$ is the ratio of the outer and inner radii of the annular region. Our focus is the long-term behavior of solutions around the steady 2D TC flow. While the laminar solution is known to be a global attractor for 2D channel flows and plane flows, it is unclear whether this is still true for rotating flows with curved geometries. In this article, we prove that the 2D Taylor-Couette flow is asymptotically stable, even at high Reynolds number ($Re\sim \nu^{-1}$), with a sharp exponential decay rate of $\exp(-\nu^{\frac13}|B|^{\frac23}R^{-2}t)$ as long as the initial perturbation is less than or equal to $\nu^\frac12 |B|^{\frac12}R^{-2}$ in Sobolev space. The powers of $\nu$ and $B$ in this decay estimate are optimal. It is derived using the method of resolvent estimates and is commonly recognized as the enhanced dissipative effect. Compared to the Couette flow, the enhanced dissipation of the rotating Taylor-Couette flow not only depends on the Reynolds number but also reflects the rotational aspect via the rotational coefficient $B$. The larger the $|B|$, the faster the long-time dissipation takes effect. We also conduct space-time estimates describing inviscid-damping mechanism in our proof. To obtain these inviscid-damping estimates, we find and construct a new set of explicit orthonormal basis of the weighted eigenfunctions for the Laplace operators corresponding to the circular flows. These provide new insights into the mathematical understanding of the 2D Taylor-Couette flows.

Abstract: In this paper, we are interested in studying the multiplicity, uniqueness, and nonexistence of solutions for a class of singular elliptic eigenvalue problem for the Dirichlet fractional $(p,q)$-Laplacian. The nonlinearity considered involves supercritical Sobolev growth. Our approach is variational togheter with the sub- and supesolution methods, and in this way we can address a wide range of problems not yet contained in the literature. Even when $W^{s_1,p}_0(\Omega) \hookrightarrow L^{\infty}\left(\Omega\right)$ failing, we establish $\|u\|_{L^{\infty}\left(\Omega\right)} \leq C[u]_{s_1,p}$ (for some $C>0$ ), when $u$ is a solution.