Analysis of PDEs (math.AP)

Abstract: This paper is concerned with the existence of transition fronts for a one-dimensional twopatch model with KPP reaction terms. Density and flux conditions are imposed at the interface between the two patches. We first construct a pair of suitable super-and subsolutions by making full use of information of the leading edges of two KPP fronts and gluing them through the interface conditions. Then, an entire solution obtained thanks to a limiting argument is shown to be a transition front moving from one patch to the other one. This propagating solution admits asymptotic past and future speeds, and it connects two different fronts, each associated with one of the two patches. The paper thus provides the first example of a transition front for a KPP-type two-patch model with interface conditions.

Abstract: We consider the Cauchy problems in the whole space for wave equations with higher derivative terms. We derive sharp growth estimates of the $L^2$-norm of the solution itself in the case of the space 1, 2 dimensions. By imposing the weighted $L^1$-initial velocity, we can get the lower and upper bound estimates of the solution itself. In three or more dimensions, we observe that the $L^2$-growth behavior of the solution never occurs in the ($L^2 \cap L^1$)-framework of the initial data.

Abstract: An initial-boundary value problem with one boundary condition is considered for the higher order nonlinear Schr\"odinger equation. It is assumed that either the boundary condition is homogeneous or the nonlinearity in the equation is quadratic. Results on existence, uniqueness and continuous dependence on input data of global weak solutions are obtained.

Abstract: We study homogenization problem for non-autonomous parabolic equations of the form $\partial_t u=L(t)u$ with an integral convolution type operator $L(t)$ that has a non-symmetric jump kernel which is periodic in spatial variables and stationary random in time. We show that the homogenized equation is a SPDE with a finite dimensional multiplicative noise.

Abstract: In this article, we investigate the gyrokinetic limit for the two-dimensional Vlasov-Poisson system with point charges. We show that the solution converges to a measure-valued solution of the Euler equation with a defect measure, which extends the results in [Miot, Nonlinearity, 2019] to the case of multi-point charges and removes the small condition $\sup_{0<\varepsilon<1}\|f_{\varepsilon}^0\|_{L^1}<1$.

Abstract: We use the framework of the first-order differential structure in metric measure spaces introduced by Gigli to define a notion of weak solutions to gradient flows of convex, lower semicontinuous and coercive functionals. We prove their existence and uniqueness and show that they are also variational solutions; in particular, this is an existence result for variational solutions. Then, we apply this technique in the case of a gradient flow of a functional with inhomogeneous growth.

Abstract: We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schr\"odinger equation $ i\partial_tu+\Delta u+\mu|x|^{-b}|u|^{\alpha}u+iau=0$, $\mu \in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re \textit{e}(a) \geq 0$, $\alpha>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a<a_*$ then blow up occurs, while for $a>a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$

Abstract: We prove the existence and uniqueness of weak solutions to a class of anisotropic elliptic equations with coefficients of convection term belonging to some suitable Marcinkiewicz spaces. Some useful a priori estimates and regularity results are also derived.

Abstract: We provide necessary and sufficient conditions on the density $W:\mathbb R^d\times\mathbb R ^d\to\mathbb R$ in order to ensure the sequential weak* lower semicontinuity of the functional $J: W^{1,\infty}(I;\mathbb R^d)\to \mathbb R$, defined as \begin{align*} J(u):=ess\,sup_{I\times I}W(u'(x), u'(y)), \end{align*} when $I$ is an open and bounded interval of $\mathbb R$. We also show that, when $d=1$, the lower semicontinuous envelope of $I$ in general can be obtained by replacing $W$ by its separately level convex envelope.

Abstract: A new notion of displacement convexity on a matrix level is developed for density flows arising from mean-field games, compressible Euler equations, entropic interpolation, and semi-classical limits of non-linear Schr\"odinger equations. Matrix displacement convexity is stronger than the classical notions of displacement convexity, and its verification (formal and rigorous) relies on matrix differential inequalities along the density flows. The matrical nature of these differential inequalities upgrades dimensional functional inequalities to their intrinsic dimensional counterparts, thus improving on many classical results. Applications include turnpike properties, evolution variational inequalities, and entropy growth bounds, which capture the behavior of the density flows along different directions in space.

Abstract: We show that the locations where finite- and infinite-time clustering occurs for the 1D Euler-alignment system can be determined using only the initial data. Our present work provides the first results on the structure of the finite-time singularity set and asymptotic clusters associated to a weak solution. In many cases, the eventual size of the cluster can be read off directly from the flux associated to a scalar balance law formulation of the system.

Abstract: We show that the Poincar\'{e} inequality holds on an open set $D\subset\mathbb{R}^n$ if and only if $D$ admits a smooth, bounded function whose Laplacian has a positive lower bound on $D$. Moreover, we prove that the existence of such a bounded, strictly subharmonic function on $D$ is equivalent to the finiteness of the strict inradius of $D$ measured with respect to the Newtonian capacity. We also obtain a sharp upper bound, in terms of this notion of inradius, for the smallest eigenvalue of the Dirichlet--Laplacian.