Analysis of PDEs (math.AP)

Abstract: The main purpose of this work is to provide an existence and uniqueness result for the solution of a linear parabolic system posed on a star-shaped network, which presents a new type of Kirchhoff's boundary transmission condition at the junction. This new type of Kirchhoff's condition-that we decide to call here local-time Kirchhoff 's condition-induces a dynamical behavior with respect to an external variable that may be interpreted as a local time parameter, designed to drive the system only at the singular point of the network. The seeds of this study point towards a forthcoming theoretical inquiry of a particular generalization of Walsh's random spider motions, whose spinning measures would select the available directions according to the local time of the motion at the junction of the network.

Abstract: The aim of this paper is to study the null controllability of a large class of quasilinear parabolic equations. In a first step we prove that the associated linear parabolic equations with non-constant diffusion coefficients are approximately null controllable by the means of regular controls and that these controls depend continuously to the diffusion coefficient. A fixed-point strategy is employed in order to prove the null approximate controllability for the considered quasilinear parabolic equations. We also show the exact null controllability in arbitrary small time for a class of parabolic equations including the parabolic $p$-Laplacian with $\frac{3}{2} < p < 2$. The theoretical results are numerically illustrated combining a fixed point algorithm and a reformulation of the controllability problem for linear parabolic equation as a mixed-formulation which is numerically solved using a finite elements method.

Abstract: This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the H{\"o}lder rates for the apriori and aposteriori error estimates obtained in this paper can exceed the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.

Abstract: The present paper is devoted to the proof of time decay estimates for derivatives at any order of finite energy global solutions of the Navier-Stokes equations in general two-dimensional domains. These estimates only depend on the order of derivation and on the L2 norm of the initial data. The same elementary method just based on energy estimates and Ladyzhenskaya inequality also leads to Gevrey regularity results.

Abstract: The goal of this article is to obtain observability estimates for Schr{\"o}dinger equations in the plane R 2. More precisely, considering a 2$\pi$Z 2-periodic potential V $\in$ L $\infty$ (R 2), we prove that the evolution equation i$\partial$tu = --$\Delta$u + V (x)u, is observable from any 2$\pi$Z 2-periodic measurable set, in any small time T > 0. We then extend Ta{\"u}ffer's recent result [T{\"a}u22] in the two-dimensional case to less regular observable sets and general bounded periodic potentials. The methodology of the proof is based on the use of the Floquet-Bloch transform, Strichartz estimates and semiclassical defect measures for the obtention of observability inequalities for a family of Schr{\"o}dinger equations posed on the torus R 2 /2$\pi$Z 2 .

Abstract: In this paper we study the inverse Laplace transform. We first derive a new global logarithmic stability estimate that shows that the inversion is severely ill-posed. Then we propose a regularization method to compute the inverse Laplace transform using the concept of mollification. Taking into account the exponential instability we derive a criterion for selection of the regularization parameter. We show that by taking the optimal value of this parameter we improve significantly the convergence of the method. Finally, making use of the holomorphic extension of the Laplace transform, we suggest a new PDEs based numerical method for the computation of the solution. The effectiveness of the proposed regularization method is demonstrated through several numerical examples.

Abstract: Let $f\in W^{3,1}_{\mathrm{loc}}(\Omega)$ be a function defined on a connected open subset $\Omega\subseteq\mathbb R^2$. We will show that its graph is contained in a quadratic surface if and only if $f$ is a weak solution to a certain system of \nth{3} order partial differential equations unless the Hessian determinant of $f$ is non-positive on the whole $\Omega$. Moreover, we will prove that the system is in some sense the simplest possible in a wide class of differential equations, which will lead to the classification of all polynomial partial differential equations satisfied by parametrizations of generic quadratic surfaces. Although we will mainly use the tools of linear and commutative algebra, the theorem itself is also somehow related to holomorphic functions.

Abstract: We prove the stability of three-dimensional axisymmetric solutions to the steady Euler system with transonic shocks in divergent nozzles under perturbations of the exit pressure and the supersonic solution in the upstream region. We first derive a free boundary problem with the newly introduced formulation of the Euler system for three-dimensional axisymmetric flows in divergent nozzles via the method of Helmholtz decomposition. We then construct an iteration scheme and use the Schauder fixed point theorem and weak implicit function theorem to solve the problem.

Abstract: This note is devoted to prove the following results on RCD(K,N)-spaces: 1) minimizers of one-phase Bernoulli problems are locally Lipschitz continuous; 2) minimizers of classical obstacle problems are quadratic growth away from the free boundary. Recently, both of these two results were obtained on non-collapsed RCD(K,N)-spaces; see [13,23]. This note will prove these two results without assuming that the ambient space is non-collapsed. We also include a proof of nondegeneracy of minimizers and locally finiteness of perimeter of their free boundaries for two-phase Bernoulli problems.

Abstract: We establish the global $C^{1, \alpha}$-regularity for functions in solution classes, whenever ellipticity constants are sufficiently close. As an application, we derive the global regularity result concerning the parabolic normalized $p$-Laplace equations, provided that $p$ is close to 2. Our analysis relies on the compactness argument with the iteration procedure.

Abstract: We consider the wave equation with focusing power nonlinearity. The associated ODE in time gives rise to a self-similar solution known as the ODE blowup. We prove the nonlinear asymptotic stability of this blowup mechanism outside of radial symmetry in all space dimensions and for all superlinear powers. This result covers for the first time the whole energy-supercritical range without symmetry restrictions.

Abstract: We provide a quantitative version of a result due to Poffald and Reich on the asymptotic behavior of solutions of a second-order Cauchy problem generated by an accretive operator in the form of a rate of convergence. This quantitative result is then used to generalize a result of Xu on the asymptotic behavior of almost-orbits of the solution semigroup of a first-order Cauchy problem to this second-order case.

Abstract: We study the uniqueness and the radial symmetry of minimizers on a Pohozaev-Nehari manifold to the Schr\"{o}dinger Poisson equation with a general nonlinearity $f(u)$. Particularly, we allow that $f$ is $L^2$ supercritical. The main result shows that minimizers are unique and radially symmetric modulo suitable translations.

Abstract: We study the existence of normalized solutions to the following logarithmic Schr\"{o}dinger equation \begin{equation*}\label{eqs01} -\Delta u+\lambda u=\alpha u\log u^2+\mu|u|^{p-2}u, \ \ x\in\R^N, \end{equation*} under the mass constraint \[ \int_{\R^N}u^2\mathrm{d}x=c^2, \] where $\alpha,\mu\in \R$, $N\ge 2$, $p>2$, $c>0$ is a constant, and $\lambda\!\in\!\R$ appears as Lagrange multiplier. Under different assumptions on $\alpha,\mu,p$ and $c$, we prove the existence of ground state solution and excited state solution. The asymptotic behavior of the ground state solution as $\mu\to 0$ is also investigated. Our results including the case $\alpha<0$ or $\mu<0$, which is less studied in the literature.

Abstract: We establish certain maximum principles for a class of strongly coupled elliptic (or cross diffusion) systems of $m\ge2$ equations. The reaction parts can be non cooperative. These new results will be crucial in obtaining coexistence and persistence for many models with cross diffusion effects.

Abstract: We derive rigorously the reduced dynamical laws for quantized vortex dynamics of the complex Ginzburg-Landau equation on torus when the core size of vortex $\varepsilon\to 0$. The reduced dynamical laws of the complex Ginzburg-Landau equation are governed by a mixed flow of gradient flow and Hamiltonian flow which are both driven by a renormalized energy on torus. Finally, some first integrals and analytic solutions of the reduced dynamical laws are discussed.

Abstract: This paper is devoted to investigating the rotating Boussinesq equations of inviscid, incompressible flows with both fast Rossby waves and fast internal gravity waves. The main objective is to establish a rigorous derivation and justification of a new generalized quasi-geostrophic approximation in a channel domain with no normal flow at the upper and lower solid boundaries, taking into account the resonance terms due to the fast and slow waves interactions. Under these circumstances, We are able to obtain uniform estimates and compactness without the requirement of either well-prepared initial data (as in [10]) or domain with no boundary (as in [17]). In particular, the nonlinear resonances and the new limit system, which takes into account the fast waves correction to the slow waves dynamics, are also identified without introducing Fourier series expansion. The key ingredient includes the introduction of (full) generalized potential vorticity.

Abstract: In this paper, we are concerned with bilinear estimates related to the two-dimensional stationary Navier--Stokes equation. By establishing concrete counter examples, we prove the bilinear estimates fail for almost all scaling critical Besov spaces. Our result may be closely related to an open problem whether the two-dimensional stationary Navier--Stokes equation on the whole plane $\mathbb{R}^2$ is well-posed or ill-posed in scaling critical Besov spaces.

Abstract: We consider the two-dimensional stationary Navier--Stokes equation on the whole plane $\mathbb{R}^2$. For the higher-dimensional cases $\mathbb{R}^n$ with $n \geqslant 3$, the stationary Navier--Stokes equation is well-posed in the scaling critical Besov spaces based on $L^p(\mathbb{R}^n)$ for $1 \leqslant p < n$ but ill-posed for $n \leqslant p \leqslant \infty$. However, the corresponding problem in the two-dimensional case $n=2$ has remained open for a long time. In the present paper, we address this open problem and prove that the stationary Navier--Stokes equation on $\mathbb{R}^2$ is ill-posed in the scaling critical Besov spaces based on $L^p(\mathbb{R}^2)$ for all $1 \leqslant p \leqslant 2$. To overcome the inherent difficulty arising in the two-dimensional analysis, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier--Stokes equation and shows that it is possible to establish weird nonstationary solutions if we suppose to contrary that the stationary problem is well-posed.

Abstract: We present a singular value polyconvex conjugation. Employing this conjugation, we derive a necessary and sufficient criterion for polyconvexity of isotropic functions by means of the convexity of a function with respect to the signed singular values. Moreover, we present a new criterion for polyconvexity of isotropic functions by means of matrix invariants.

Abstract: In this paper, we consider the homogeneous complex k-Hessian equation in $\Omega\backslash\{0\}$. We prove the existence and uniqueness of the $C^{1,\alpha}$ solution by constructing approximating solutions. The key point for us is to construct the subsolution for approximating problem and establish uniform gradient estimates and complex Hessian estimates which is independent of the approximation.

Abstract: We consider a shape optimization problem related to the persistence threshold for a biological species, the unknown shape corresponding to the zone of the habitat which is favorable to the population. Analytically, this translates in the minimization of a weighted eigenvalue of the Dirichlet Laplacian, with respect to a bang-bang indefinite weight. For such problem, we provide a full description of the singularly perturbed regime in which the volume of the favorable zone vanishes, with particular attention to the interplay between its location and shape. First, we show that the optimal favorable zone shrinks to a connected, nearly spherical set, in $C^{1,1}$ sense, which aims at maximizing its distance from the lethal boundary. Secondly, we show that the spherical asymmetry of the optimal favorable zone decays exponentially, with respect to a negative power of its volume, in the $C^{1,\alpha}$ sense, for every $\alpha<1$. This latter property is based on sharp quantitative asymmetry estimates for the optimization of a weighted eigenvalue problem on the full space, of independent interest.

Abstract: We study the insulated conductivity problem with closely spaced insulators embedded in a homogeneous matrix where the current-electric field relation is the power law $J = |E|^{p-2}E$. The gradient of solutions may blow up as $\varepsilon$, the distance between insulators, approaches to 0. In 2D, we prove an upper bound of the gradient to be of order $\varepsilon^{-\alpha}$, where $\alpha = 1/2$ when $p \in(1,3]$ and any $\alpha > 1/(p-1)$ when $p > 3$. We provide examples to show that this exponent is almost optimal. In dimensions $n \ge 3$, we prove an upper bound of order $\varepsilon^{-1/2 + \beta}$ for some $\beta > 0$, and show that $\beta \nearrow 1/2$ as $n \to \infty$.