Analysis of PDEs (math.AP)

Abstract: In this paper, we study weighted fractional Sobolev-Poincar\'e inequalities for irregular domains. The weights considered here are distances to the boundary to certain powers, and the domains are the so-called $s$-John domains and $\beta$-H\"older domains. Our main results extend that of Hajlasz-Koskela [J. Lond. Math. Soc. 1998] from the classical weighted Sobolev-Poincar\'e inequality to its fractional counter-part and Guo [Chin. Ann. Math. 2017] from the frational Sobolev-Poincar\'e inequality to its weighted case.

Abstract: We are concerned with a one dimensional flow of a compressible fluid which may be seen as a simplification of the flow of fluid in a long thin pipe. We assume that the pipe is on one side ended by a spring. The other side of the pipe is let open -- there we assume either inflow or outflow boundary conditions. Such situation can be understood as a toy model for human lungs. We tackle the question of uniqueness and existence of a strong solution for a system modelling the above process, special emphasis is laid upon the estimate of the maximal time of existence.

Abstract: We show that the recent work by G{\'e}rard-Kappeler-Topalov can be used in order to construct new non degenerate invariant measures for the Benjamin-Ono equation on the Sobolev spaces H s , s > --1/2.

Abstract: Let $\Delta_k$ be the Dunkl Laplacian relative to a fixed root system $\mathcal{R}$ in $\mathbb{R}^d$, $d\geq2$, and to a nonnegative multiplicity function $k$ on $\mathcal{R}$. Our first purpose in this paper is to solve the $\Delta_k$-Dirichlet problem for annular regions. Secondly, we introduce and study the $\Delta_k$-Green function of the annulus and we prove that it can be expressed by means of $\Delta_k$-spherical harmonics. As applications, we obtain a Poisson-Jensen formula for $\Delta_k$-subharmonic functions and we study positive continuous solutions for a $\Delta_k$-semilinear problem.

Abstract: In this work, we suggest a system of differential equations that quantitatively models the formulation and evolution of a trend cycle through the consideration of underlying dynamics between the trend participants. Our model captures the five stages of a trend cycle, namely, the onset, rise, peak, decline, and obsolescence. It also provides a unified mathematical criterion/condition to characterize the fad, fashion and classic. We prove that the solution of our model can capture various trend cycles. Numerical simulations are provided to show the expressive power of our model.

Abstract: Of concern are lump solutions for the fractional Kadomtsev--Petviashvili (fKP) equation. As in the classical Kadomtsev--Petviashvili equation, the fKP equation comes in two versions: fKP-I (strong surface tension case) and fKP-II (weak surface tension case). We prove the existence of nontrivial lump solutions for the fKP-I equation in the energy subcritical case $\alpha>\frac{4}{5}$ by means of variational methods. It is already known that there exist neither nontrivial lump solutions belonging to the energy space for the fKP-II equation nor for the fKP-I when $\alpha \leq \frac{4}{5}$. Furthermore, we show that for any $\alpha>\frac{4}{5}$ lump solutions for the fKP-I equation are smooth and decay quadratically at infinity. Numerical experiments are performed for the existence of lump solutions and their decay. Moreover, numerically, we observe cross-sectional symmetry of lump solutions for the fKP-I equation.

Abstract: In this paper, we consider a semi-classical version of the nonhomogeneous heat equation with singular time-dependent coefficients on the lattice $\hbar \mathbb{Z}^n$. We establish the well-posedeness of such Cauchy equations in the classical sense when regular coefficients are considered, and analyse how the notion of very weak solution adapts in such equations when distributional coefficients are regarded. We prove the well-posedness of both the classical and the very weak solution in the weighted spaces $\ell^{2}_{s}(\hbar \mathbb{Z}^n)$, $s \in \mathbb{R}$, which is enough to prove the well-posedness in the space of tempered distributions $\mathcal{S}'(\hbar \mathbb{Z}^n)$. Notably, when $s=0$, we show that for $\hbar \rightarrow 0$, the classical (resp. very weak) solution of the heat equation in the Euclidean setting $\mathbb{R}^n$ is recaptured by the classical (resp. very weak) solution of it in the semi-classical setting $\hbar \mathbb{Z}^n$.

Abstract: In this note we revisit a result in [9], where we established nonlocal isoperimetric inequalities and the related embeddings for Besov spaces adapted to a class of H\"ormander operators of Kolmogorov-type. We provide here a new proof which exploits a weak-type Sobolev embedding established in [11].

Abstract: In this note, we prove an existence result for generalized Kazdan-Warner equations on compact Riemannian manifolds by using the flow approach or the upper and lower solution method. In addition, we give a prior estimate for this type equations.

Abstract: We prove the unique solvability for the Poisson and heat equations in non-smooth domains $\Omega\subset \mathbb{R}^d$ in weighted Sobolev spaces. The zero Dirichlet boundary condition is considered, and domains are merely assumed to admit the Hardy inequality: $$ \int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\,\,\mathrm{d} x\leq N\int_{\Omega}|\nabla f|^2 \,\mathrm{d} x\,\,\,\,,\,\,\,\, \forall f\in C_c^{\infty}(\Omega)\,. $$ To describe the boundary behavior of solutions, we introduce a weight system that consists of superharmonic functions and the distance function to the boundary. The results provide separate applications for the following domains: convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, conic domains, and domains $\Omega\subset\mathbb{R}^d$ which the Aikawa dimension of $\Omega^c$ is less than $d-2$.

Abstract: We introduce new techniques to study the differential complexes associated to tube structures on $M \times \mathbb{T}^m$ of corank $m$, in which $M$ is a compact manifold and $\mathbb{T}^m$ is the $m$-torus. By systematically employing partial Fourier series, for complex tube structures, we completely characterize global solvability, in a given degree, in terms of a weak form of hypoellipticity, thus generalizing existing results and providing a broad answer to an open problem proposed by Hounie and Zugliani (2017). We also obtain new results on the finiteness of the cohomology spaces in intermediate degrees. In the case of real tube structures, we extend an isomorphism for the cohomology spaces originally obtained by Dattori da Silva and Meziani (2016) in the case $M = \mathbb{T}^n$. Moreover, we establish necessary and sufficient conditions for the differential operator to have closed range in the first degree.

Abstract: We propose an approximate model for the 2D Kuramoto-Sivashinsky equations (KSE) of flame fronts and crystal growth. We prove that this new ``calmed'' version of the KSE is globally well-posed, and moreover, its solutions converge to solutions of the KSE on the time interval of existence and uniqueness of the KSE at an algebraic rate. In addition, we provide simulations of the calmed KSE, illuminating its dynamics. These simulations also indicate that our analytical predictions of the convergence rates are sharp. We also discuss analogies with the 3D Navier-Stokes equations of fluid dynamics.

Abstract: This paper provides a para-differential calculus toolbox on compact Lie groups and homogeneous spaces. It helps to understand non-local, nonlinear partial differential operators with low regularity on manifolds with high symmetry. In particular, the paper provides a para-linearization formula for the Dirichlet-Neumann operator of a distorted 2-sphere, a key ingredient in understanding long-time behaviour of spherical capillary water waves. As an initial application, the paper provides a new proof of local well-posedness for spherical capillary water waves equation under weaker regularity assumptions compared to previous results.

Abstract: Inspired by a planar partitioning problem involving multiple unbounded chambers, using classical techniques this article investigates what can be said of the existence, uniqueness, and regularity of minimizers in a certain free-endpoint isoperimetric problem. By restricting to curves which are expressible as graphs of functions, a full existence-uniqueness-regularity result is proved using a convexity technique inspired by work of Talenti. The problem studied here can be interpreted physically as the identification of the equilibrium shape of a sessile liquid drop in half-space (in the absence of gravity). This is a well-studied variational problem whose full resolution requires the use of geometric measure theory, in particular the theory of sets of finite perimeter, but here we present a more direct, classical geometrical approach.