Journal of Evolution Equations

Bignamini, Davide A.; De Fazio, Paolo

doi: 10.1007/s00028-024-01005-1pmid: N/A

In an infinite-dimensional separable Hilbert space X, we study the realizations of Ornstein–Uhlenbeck evolution operators Ps,t\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{s,t}$$\end{document} in the spaces Lp(X,γt)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L^p(X,\gamma _t)$$\end{document}, {γt}t∈R\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\{\gamma _t\}_{t\in \mathbb {R}}$$\end{document} being a suitable evolution system of measures for Ps,t\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$P_{s,t}$$\end{document}. We prove hypercontractivity results, relying on suitable Log-Sobolev estimates. Among the examples, we consider the transition evolution operator associated with a non-autonomous stochastic parabolic PDE.

Golse, François; Imbert, Cyril; Ji, Sehyun; Vasseur, Alexis F.

doi: 10.1007/s00028-024-01009-xpmid: N/A

This paper deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix aij(z)=|z|γ(|z|2δij-zizj)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a_{ij} (z)=|z|^\gamma (|z|^2 \delta _{ij} - z_iz_j)$$\end{document} for γ∈[-3,-2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma \in [-3,-2)$$\end{document}. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in Villani (Arch Rational Mech Anal 143:273–307, 1998) has zero Pm∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathscr {P}^{m_*}$$\end{document} parabolic Hausdorff measure with m∗:=72|2+γ|\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m_*:= \frac{7}{2} |2+\gamma |$$\end{document}. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.

Ploß, David

doi: 10.1007/s00028-024-01015-zpmid: N/A

For bounded domains Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega $$\end{document} with Lipschitz boundary Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}, we investigate boundary value problems for elliptic operators with variable coefficients of fourth-order subject to Wentzell (or dynamic) boundary conditions. Using form methods, we begin by showing general results for an even wider class of operators of type A=B∗B0-NbBγ,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} {\mathcal {A}}=\begin{pmatrix} B^*B & 0 \\ -{\mathscr {N}}^{\mathfrak {b}}B & \gamma \end{pmatrix}, \end{aligned}$$\end{document}where B is associated to a quadratic form b\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathfrak {b}}$$\end{document} and Nb\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathscr {N}}^{{\mathfrak {b}}}$$\end{document} an abstractly defined co-normal Neumann trace. Even in this general setting, we prove generation of an analytic semigroup on the product space H:=L2(Ω)×L2(Γ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {H}}:=L^2(\Omega ) \times L^2(\Gamma )$$\end{document}. Using recent results concerning weak co-normal traces, we apply our abstract theory to the elliptic fourth-order case and are able to fully characterize the domain in terms of Sobolev regularity for operators in divergence form B=-divQ∇\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B=-\mathop {{div} }Q \nabla $$\end{document} with Q∈C1,1(Ω¯,Rd×d),\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q \in C^{1,1}({\overline{\Omega }},{\mathbb {R}}^{d\times d}),$$\end{document} also obtaining Hölder-regularity of solutions. Finally, we also discuss asymptotic behavior and (eventual) positivity.

Jena, Vaibhav Kumar; Maity, Debayan; Sufian, Abu

doi: 10.1007/s00028-024-01006-0pmid: N/A

We study a free boundary value problem modelling the motion of a piston in a viscous compressible fluid. The fluid is modelled by 1D compressible Navier–Stokes equations with possibly degenerate viscosity coefficient, and the motion of the piston is described by Newton’s second law. We show that the initial boundary value problem has a unique global in time solution, and we also determine the large time behaviour of the system. Finally, we show how our methodology may be adapted to the motion of several pistons.

Mukherjee, Arpan; Sini, Mourad

doi: 10.1007/s00028-024-01004-2pmid: N/A

We analyse the ultrasound waves reflected by multiple bubbles in the linearized time-dependent acoustic model. The generated time-dependent wave field is estimated close to the bubbles. The motivation of this study comes from the therapy modality using acoustic cavitation generated by injected bubbles into the region of interest. The goal is to create enough, but not too much, pressure in the region of interest to eradicate anomalies in that region. In a previous work, we already showed that, in case of single bubble, the dominant part of the acoustic pressure near it splits into two main echoes. The primary one is the incident field shifted and amplified at certain order. The secondary one is of periodic form which is related to the resonant frequency (i.e. the Minnaert one) created by the single bubble. This secondary wave can be amplified at will, at certain specific times, by tuning properly the material characteristics of the used bubble. Here, we derive the dominant part of the generated acoustic field by a cluster of bubbles taking into account the (high) contrasts of their mass density and bulk as well as their general distribution in the given region. As consequences of these approximations, we highlight the following features: If we use dimers (two close bubbles), or generally polymers, then both the primary and the secondary waves can be amplified resulting in a remarkable enhancement of the whole echo in the whole time. The main reason for that is the closeness of the bubbles which translates the fact that the polymers resonate (even if each bubble do not). This feature is shown also when we use a set of separated polymers. Therefore, one can generate desired amount of pressure by injecting such a set of polymers.If we distribute the bubbles everywhere in the region of interest, in a periodic way for instance, then we can derive the effective acoustic model which turns out to be a dispersive one (due to the resonant behaviour of the bubbles). Therefore, the original question of generating desired acoustic pressure can be related to the effective model. We show that for a given desired pressure, we can tune the effective model to generate it.

Oka, Yusuke; Zhanpeisov, Erbol

doi: 10.1007/s00028-024-01025-xpmid: N/A

We consider the Cauchy problem for a time fractional semilinear heat equation [graphic not available: see fulltext]where 0<α<1,γ>1,N∈Z⩾1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\alpha <1,\, \gamma >1,\, N\in \mathbb {Z}_{\geqslant 1}$$\end{document} and μ(x)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mu (x)$$\end{document} belongs to inhomogeneous/homogeneous Besov–Morrey spaces. The fractional derivative C∂tα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$^{C}\partial ^{\alpha }_{t}$$\end{document} is interpreted in the Caputo sense. We present sufficient conditions for the existence of local/global-in-time solutions to problem (P). Our results cover all existing results in the literature and can be applied to a large class of initial data.

Lou, Guangpu; Dong, Jianwei

doi: 10.1007/s00028-024-01019-9pmid: N/A

In this paper, we study the analytical solutions to the pressureless Euler–Korteweg equations. First, we construct a self-similar solution to the one-dimensional free boundary value problem with vacuum boundary condition and show that the free boundary spreads out linearly in time. Second, we extend this result to the N-dimensional radially symmetric case and the three-dimensional cylindrically symmetric case, respectively.

Lin, Qiang; Zhang, Binlin

doi: 10.1007/s00028-024-01008-ypmid: N/A

In this paper, we show a blowup criterion of solution for a parabolic equation with critical exponential source and arbitrary positive initial energy, which generalizes the blowup conclusions in reference (Ishiwata et al. in J Evol Equ 21:1677–1716, 2021) for subcritical and critical initial energy cases that depend on the depth of the potential well. Additionally, the continuous dependence of the local solution on the initial data is proved in detail.

Djilali, Salih; Bentout, Soufiane; Tridane, Abdessamad

doi: 10.1007/s00028-024-01013-1pmid: N/A

This paper explores a generalized nonlocal dispersion SIS epidemic model subject to the Neumann boundary conditions and spatial heterogeneity. We use a convolution operator to describe the nonlocal spatial movements of individuals. Our primary goal is to investigate this model, focusing on a generalized incidence function, which presents an additional challenge in the model analysis. This model’s basic reproduction number, R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_0$$\end{document}, is identified, and it is proved that 1-R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$1 - R_0$$\end{document} has the same sign as the principal eigenvalue of a generalized linear nonlocal operator. Furthermore, the asymptotic profiles of R0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_0$$\end{document} in terms of dispersion coefficients are also established. We also investigate the existence and uniqueness of an endemic steady state for R0>1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$R_0 > 1$$\end{document}, and we study the large dispersal rates effect on the asymptotic profiles of the steady endemic state. Finally, we discussed the global asymptotic behavior of the solution for different dispersal coefficients.

Gess, Benjamin; Röckner, Michael; Wu, Weina

doi: 10.1007/s00028-024-01023-zpmid: N/A

We establish a framework for the existence and uniqueness of solutions to stochastic nonlinear (possibly multi-valued) diffusion equations driven by multiplicative noise, with the drift operator L being the generator of a transient Dirichlet form on a finite measure space (E,B,μ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(E,\mathcal {B},\mu )$$\end{document} and the initial value in Fe∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {F}_e^*$$\end{document}, which is the dual space of an extended transient Dirichlet space. L and Fe∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathcal {F}_e^*$$\end{document} replace the Laplace operator Δ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta $$\end{document} and H-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H^{-1}$$\end{document}, respectively, in the classical case. This framework includes stochastic fast diffusion equations, stochastic fractional fast diffusion equations, the Zhang model, and applies to cases with E being a manifold, a fractal, or a graph. In addition, our results apply to operators -f(-L)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-f(-L)$$\end{document}, where f is a Bernstein function, e.g., f(λ)=λα\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(\lambda )=\lambda ^\alpha $$\end{document} or f(λ)=(λ+1)α-1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$f(\lambda )=(\lambda +1)^\alpha -1$$\end{document}, 0<α<1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$0<\alpha <1$$\end{document}.