Analysis of PDEs (math.AP)

Abstract: In this paper we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schr\"odinger equations of the type $$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $\Omega$},$$ where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$. In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.

Abstract: This paper concerns the sharp interface limit of solutions to the inhomogeneous incompressible Navier-Stokes/Allen-Cahn coupled system in a bounded domain $\Omega \subset \mathbb{R}^n,\ n =2,3$. Based on a relative energy method, we prove that the solutions to the Navier-Stokes/Allen-Cahn system converge to the corresponding solutions to a sharp interface model provided that the thickness of the diffuse interfacial zone goes to zero. It is noted that the relative energy method can avoid both the spectral estimates of the linearized Allen-Cahn operator and the construction of approximate solutions by the matched asymptotic expansion method in the study of the sharp interface limit process. And some suitable functionals are designed and estimated by elaborated energy methods accordingly.

Abstract: We provide an overview of the results on Hughes' model for pedestrian movements available in the literature. After the first successful approaches to solving a regularised version of the model, researchers focused on the structure of the Riemann problem, which led to local-in-time existence results for Riemann-type data and paved the way for a WFT (Wave-Front Tracking) approach to the solution semigroup. In parallel, a DPA (Deterministic Particles Approximation) approach was developed in the spirit of follow-the-leader approximation results for scalar conservation laws. Beyond having proved to be powerful analytical tools, the WFT and the DPA approaches also led to interesting numerical results. However, only existence theorems on very specific classes of initial data (essentially ruling out non-classical shocks) have been available until very recently. A proper existence result using a DPA approach was proven not long ago in the case of a linear coupling with the density in the eikonal equation. Shortly after, a similar result was proven via a fixed point approach. We provide a detailed statement of the aforementioned results and sketch the main proofs. We also provide a brief overview of results that are related to Hughes' model, such as the derivation of a dynamic version of the model via a mean-field game strategy, an alternative optimal control approach, and a localized version of the model. We also present the main numerical results within the WFT and DPA frameworks.

Abstract: We are interested in the stability analysis of two-dimensional incompressible inviscid fluids. Specifically, we revisit a recent result on the stability of Yudovich's solutions to the incompressible Euler equations in $L^\infty([0,T];H^1)$ by providing a new approach to its proof based on the idea of compactness extrapolation and by extending it to the whole plane. This new method of proof is robust and, when applied to viscous models, leads to a remarkable logarithmic improvement on the rate of convergence in the vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this logarithmic gain is the result of the fact that, in appropriate high-regularity settings, the smoothness of solutions to the Euler equations at times $t\in [0,T)$ is strictly higher than their regularity at time $t=T$. This ``memory effect'' seems to be a general principle which is not exclusive to fluid mechanics. It is therefore likely to be observed in other setting and deserves further investigation. Finally, we also apply the stability results on Euler systems to the study of two-dimensional ideal plasmas and establish their convergence, in strong topologies, to solutions of magnetohydrodynamic systems, when the speed of light tends to infinity. The crux of this asymptotic analysis relies on a fine understanding of Maxwell's system.

Abstract: We consider semilinear elliptic equations with mixed boundary conditions in spherical sectors inside a cone. The aim of the paper is to show that a radial symmetry result of Gidas-Ni-Nirenberg type for positive solutions does not hold in general nonconvex cones. This symmetry breaking result is achieved by studying the Morse index of radial positive solutions and analyzing how it depends on the domain D on the unit sphere which spans the cone. In particular it is proved that the Neumann eigenvalues of the Laplace Beltrami operator on D play a role in computing the Morse index. A similar breaking of symmetry result is obtained for the positive solutions of the critical Neumann problem in the whole unbounded cone. In this case it is proved that the standard bubbles, which are the only radial solutions, become unstable for a class of nonconvex cones.

Abstract: In this paper, we consider the compressible Euler-Poisson equations for polytropes $P(\rho)=K\rho^\gamma$ and the white dwarf star. Firstly, we develop two variational problem for $\gamma=\frac{4}{3}$ and $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$ respectively. The first variational problem for $\gamma=\frac{4}{3}$ is related to the best constant of a Hardy-Littlewood type inequality. The best constant obtained is sharp and it yields a threshold of the mass to the gaseous star which is the Chandrasekhar limit mass. For $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$, we construct a type of cross constrained variational problem attained by the Lane-Emden function. Then, we show that the spherically symmetric finite energy weak solution globally exists if the mass is less than the Chandrasekhar limit mass for $\gamma=\frac{4}{3}$ or the initial data belongs to an invariant set constructed by the cross-constrained variational argument for $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$. Furthermore, we conditionally obtain that the support of the gaseous star expands as time tends to infinity with a virial argument. We also consider the white dwarf star and prove that if the mass is less than the Chandrasekhar limit mass, the white dwarf star cannot collapse to a point.

Abstract: We construct the traveling wave solutions of an FKPP growth process of two densities of particles, and prove that the critical traveling waves are locally stable in a space where the perturbations can grow exponentially at the back of the wave. The considered reaction-diffusion system was introduced by Hannezo et al. in the context of branching morphogenesis (Cell, 171(1):242-255.e27, 2017): active, branching particles accumulate inactive particles, which do not react. Thus, the system features a continuum of steady state solutions, complicating the analysis. We adopt a result by Faye and Holzer (J.Diff.Eq., 269(9):6559-6601, 2020) for proving the stability of the critical traveling waves, by modifying the semi-group estimates to spaces with unbounded weights. The novelty is that we use a Feynman-Kac formula to get an exponential a priori estimate for the tail of the PDE. This supersedes the need for an integrable weight.

Abstract: The existence and the regularity results obtained in [37] for the variational model introduced in [36] to study the optimal shape of crystalline materials in the setting of stress-driven rearrangement instabilities (SDRI) are extended from two dimensions to any dimensions $n\geq2$. The energy is the sum of the elastic and the surface energy contributions, which cannot be decoupled, and depend on configurational pairs consisting of a set and a function that model the region occupied by the crystal and the bulk displacement field, respectively. By following the physical literature, the ``driving stress'' due to the mismatch between the ideal free-standing equilibrium lattice of the crystal with respect to adjacent materials is included in the model by considering a discontinuous mismatch strain in the elastic energy. Since two-dimensional methods and the methods used in the previous literature where Dirichlet boundary conditions instead of the mismatch strain and only the wetting regime were considered, cannot be employed in this setting, we proceed differently, by including in the analysis the dewetting regime and carefully analyzing the fine properties of energy-equibounded sequences. This analysis allows to establish both a compactness property in the family of admissible configurations and the lower-semicontinuity of the energy with respect to the topology induced by the $L^1$-convergence of sets and a.e.\ convergence of displacement fields, so that the direct method can be applied. We also prove that our arguments work as well in the setting with Dirichlet boundary conditions.

Abstract: We continue our work \cite{Glo22a} on the analysis of spatially global stability of self-similar blowup profiles for semilinear wave equations in the radial case. In this paper we study the Yang-Mills equations in $(1+d)$-dimensional Minkowski space. For $d \geq 5$, which is the energy supercritical case, we consider an explicitly known equivariant self-similar blowup solution and establish its nonlinear global-in-space asymptotic stability under small equivariant perturbations. The size of the initial data is measured in terms of, in a certain sense, optimal Sobolev norm above scaling. This result complements the existing stability results in odd dimensions, while for even dimensions it is new.

Abstract: Consider operators $L_{V}:=\Delta + V$ in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^N$. Assume that $V\in C^\alpha(\Omega)$ satisfies $|V(x)| \leq \bar a\,\mathrm{dist}(x,\partial\Omega)^{-2}$ in $\Omega$ and that $L_V$ has a (minimal) ground state $\Phi_V$ in $\Omega$. We derive a representation formula for signed supersolutions (or subsolutions) of $L_Vu=0$ possessing an $L_V$ boundary trace. We apply this formula to the study of some questions of existence and uniqueness for an associated semilinear boundary value problem with signed measure data.

Abstract: We construct stationary statistical solutions of a deterministic unforced nonlinear Schr\"odinger equation, by perturbing it by a linear damping $\gamma u$ and a stochastic force whose intensity is proportional to $\sqrt \gamma$, and then letting $\gamma\to 0^+$. We prove indeed that the family of stationary solutions $\{U_\gamma\}_{\gamma>0}$ of the perturbed equation possesses an accumulation point for any vanishing sequence $\gamma_j\to 0^+$ and this stationary limit solves the deterministic unforced nonlinear Schr\"odinger equation and is not the trivial zero solution. This technique has been introduced in [KS04], using a different dissipation. However considering a linear damping of zero order and weaker solutions we can deal with larger ranges of the nonlinearity and of the spatial dimension; moreover we consider the focusing equation and the defocusing equation as well.