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arXiv2019

Mean-Field and Classical Limit for the N-Body Quantum Dynamics with Coulomb Interaction

This paper proves the validity of the joint mean-field and classical limit of the quantum $N$-body dynamics leading to the pressureless Euler-Poisson system for factorized initial data whose first marginal has a monokinetic Wigner measure. The interaction potential is assumed to be the repulsive Coulomb potential. The validity of this derivation is limited to finite time intervals on which the Euler-Poisson system has a smooth solution that is rapidly decaying at infinity. One key ingredient in the proof is an inequality from [S. Serfaty, with an appendix of M. Duerinckx arXiv:1803.08345v3 [math.AP]].

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arXiv2021

Blow-up in a quasilinear parabolic-elliptic Keller-Segel system with logistic source

This paper deals with the quasilinear parabolic-elliptic Keller-Segel system with logistic source, \begin{align*} u_t=Δ(u+1)^m - χ\nabla \cdot (u(u+1)^{α- 1} \nabla v) + λ(|x|) u - μ(|x|) u^κ, \quad 0=Δv - v + u, \quad x\inΩ,\ t>0, \end{align*} where $Ω:=B_{R}(0)\subset\mathbb{R}^n\ (n\ge3)$ is a ball with some $R>0$; $m>0$, $χ>0$, $α>0$ and $κ\ge1$; $λ$ and $μ$ are spatially radial nonnegative functions. About this problem, Winkler (Z. Angew. Math. Phys.; 2018; 69; Art. 69, 40) found the condition for $κ$ such that solutions blow up in finite time when $m=α=1$. In the case that $m=1$ and $α\in(0,1)$ as well as $λ$ and $μ$ are constant, some conditions for $α$ and $κ$ such that blow-up occurs were obtained in a previous paper (Math. Methods Appl. Sci.; 2020; 43; 7372-7396). Moreover, in the case that $m\ge1$ and $α=1$ Black, Fuest and Lankeit (arXiv:2005.12089[math.AP]) showed that there exists initial data such that solutions blow up in finite time under some conditions for $m$ and $κ$. The purpose of the present paper is to give conditions for $m\ge1$, $α>0$ and $κ\ge1$ such that solutions blow up in finite time.

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arXiv2021

Well-posedness for a class of phase-field systems modeling prostate cancer growth with fractional operators and general nonlinearities

This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy that has been recently introduced and analyzed (see [P. Colli et al., Mathematical analysis and simulation study of a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects, Math. Models Methods Appl. Sci. 30 (2020), 1253-1295], preprint in arXiv:1907.11618 [math.AP]). The related system includes two evolutionary operator equations involving fractional powers of selfadjoint, nonnegative, unbounded linear operators having compact resolvents. Both equations contain nonlinearities and in particular the equation describing the dynamics of the tumor phase variable has the structure of a Allen-Cahn equation with double-well potential and additional nonlinearity depending also on the other variable, which represents the nutrient concentration. The equation for the nutrient concentration is nonlinear as well, with a term coupling both variables. For this system we design an existence, uniqueness and continuous dependence theory by setting up a careful analysis which allows the consideration of nonsmooth potentials and the treatment of continuous nonlinearities with general growth properties.

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arXiv2023

Determining an unbounded potential for an elliptic equation with a power type nonlinearity

In this article we focus on inverse problems for a semilinear elliptic equation. We show that a potential $q$ in $L^{n/2+\varepsilon}$, $\varepsilon>0$, can be determined from the full and partial Dirichlet-to-Neumann map. This extends the results from [M. Lassas, T. Liimatainen, Y.-H. Lin, and M. Salo, Partial data inverse problems and simultaneous recovery of boundary and coefficients for semilinear elliptic equations, Rev. Mat. Iberoam. (2021)] where this is shown for Hölder continuous potentials. Also we show that when the Dirichlet-to-Neumann map is restricted to one point on the boundary, it is possible to determine a potential $q$ in $L^{n+\varepsilon}$. The authors of arXiv:2202.05290 [math.AP] proved this to be true for Hölder continuous potentials.

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arXiv2022

Propagation of velocity moments and uniqueness for the magnetized Vlasov-Poisson system

We present two results regarding the three-dimensional Vlasov--Poisson system in the full space with an external magnetic field. First, we investigate the propagation of velocity moments for solutions to the system when the magnetic field is uniform and time-dependent. We combine the classical moment approach with an induction procedure depending on the cyclotron period $T_c=||B||_{\infty}^{-1}$. This allows us to obtain, like in the unmagnetized case, the propagation of velocity moments of order $k>2$ in the full space case and of order $k>3$ in the periodic case. Second, this time taking a general magnetic field that depends on both time and position, we manage to extend a result by Miot arXiv:1409.6988v1 [math.AP] regarding uniqueness for Vlasov--Poisson to the magnetized framework.

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arXiv2025

A quasilinear Keller-Segel model with saturated discontinuous advection

We consider the singular limit of a chemotaxis model of bacterial collective motion recently introduced in arXiv:2009.11048 [math.AP]. The equation models aggregation-diffusion phenomena with advection that is discontinuous and depends sharply on the gradient of the density itself. The quasi-linearity of the problem poses major challenges in the construction of the solution and complications arise in the proof of regularity. Our method overcomes these obstacle by relying solely on entropy inequalities and the theory of monotone operators. We provide existence, uniqueness and smoothing estimates in any dimensional space.

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arXiv2024

The effect of protest management strategies on protesting activity and social tension: a mathematical perspective

Protest activity, a constitutionally protected right in the United States under the First Amendment, serves as a key tool for individuals with limited individual influence to unite collectively and amplify their impact. Despite its legal recognition, the historical interaction between the government and protesters has been complex. In this study, we extend a model proposed in arXiv:1502.04725 [math.AP] to incorporate the influence of protest management strategies. Our research establishes the existence and stability of traveling wave solutions, supported by both theoretical analyses and numerical simulations. We delve into the impact of two distinct protest management approaches on the qualitative and semi-quantitative characteristics of these traveling waves. These metrics can aid in categorizing different types of protests.

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arXiv2025

Nonexistence Results for a General Class of Parabolic Problems with a Potential on Weighted Graphs

We establish nonexistence conditions for nonnegative nontrivial solutions to a class of semilinear parabolic equations with a positive potential on weighted graphs, extending results in arXiv:2404.12058 [math.AP] to a broader setting that includes both the Porous Medium Equation and the Fast Diffusion Equation. We identify conditions related to the graph's geometry, the potential's behaviour at infinity, and bounds on the Laplacian of the distance function under which nonexistence holds. Using a test function argument, we derive explicit parameter ranges for nonexistence.

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arXiv2025

Stability inequalities with explicit constants for a family of reverse Sobolev inequalities on the sphere

We prove a stability inequality associated to the reverse Sobolev inequality on the sphere $\mathbb S^n$, for the full admissible parameter range $s - \frac{n}{2} \in (0,1) \cup (1,2)$. To implement the classical proof of Bianchi and Egnell, we overcome the main difficulty that the underlying operator $A_{2s}$ is not positive definite. As a consequence of our analysis and recent results from Gong et al. (arXiv:2503.20350 [math.AP]), the case $s - \frac{n}{2} \in (1,2)$ remarkably constitutes the first example of a Sobolev-type stability inequality (i) whose best constant is explicit and (ii) which does not admit an optimizer.

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arXiv2025

Unconditional Uniqueness of 5th Order KP Equations

In this paper we study the $5$th Order Kadomstev-Petviashvili (KP) equations posed on the real line. In particular we adapt the energy estimate argument from Guo-Molinet (arXiv:2404.12364v1 [math.AP]) to conclude unconditional uniqueness of the solution to data map for $5$th order KP type equations. Applying short-time $X^{s,b}$ methods to improve classical energy estimates provides more than sufficient decay when considering estimates on the interior of the time interval $[0,T]$. The issue is how we deal with the boundary. By abusing symmetry we can apply multilinear interpolation to gain access to $L^4$ Strichartz estimates, which provide improved derivative gain. When taken together, the regularity of our resultant function space can be arbitrarily close to $L^2$, which in the context of unconditional uniqueness results is almost sharp.

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arXiv2006

Multilinear estimates for periodic KdV equations and applications

We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. In particular we prove a multilinear estimate which completes the proof of global well-posedness for periodic KdV in a preceding paper (math.AP/0110045) down to the optimal regularity H^{-1/2}.

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arXiv2005

Well-posedness of a multiscale model for concentrated suspensions

In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system.

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arXiv2006

Determining nonsmooth first order terms from partial boundary measurements

We extend results of Dos Santos Ferreira-Kenig-Sjoestrand-Uhlmann (math.AP/0601466) to less smooth coefficients, and we show that measurements on part of the boundary for the magnetic Schroedinger operator determine uniquely the magnetic field related to a Hoelder continuous potential. We give a similar result for determining a convection term. The proofs involve Carleman estimates, a smoothing procedure, and an extension of the Nakamura-Uhlmann pseudodifferential conjugation method to logarithmic Carleman weights.

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arXiv2006

Global well-posedness, scattering and blow-up for the energy critical focusing non-linear wave equation

We prove, for the energy critcal, focusing NLW, that for Cauchy data (u_0, u_1) whose energy is smaller than that of (W,0), where W is the well-known radial positive solution to the corresponding ellipyic equation, the following dichotomy holds: a) if the homogeneous Sobolev norm H^1 of u_0 is smaller than that of W, we have global well-posedness and scattering, b) if the homogeneous Sobolev norm H^1 of u_0 is larger than that of W, there is blow-up in finite time. Our general approach is the one we introduced in our previous work on the corresponding problem for NLS (math.AP/0610266, Inventiones Math 2006, Online First), where we proved the corresponding result for NLS in the radial case. In the case of the wave equation we are able to treat general data by using a further conservation law in the energy space, the finite speed of propagation and Lorentz transformations to establish a crucial orthogonality property for " energy critical " elements. To prove the required rigidity theorem in the case of blow-up in finite time, we cannot use the invariance of the L^2 norm as in the case of NLS. Instead, (following earlier work of Merle-Zaag and of Giga-Kohn in the parabolic case) we introduce self-similar variables. We thus find a further Liapunov function which allows us to reduce matters to a degenerate elliptic problem with critical non-linearity. We use unique continuation to rule out the existence of non-zero solutions for the degenerate elliptic problem, thus completing the proof.

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arXiv2006

One Dimensional Conformal Metric Flows

This is the second paper of our series of papers on one dimensional conformal metric flows. In this paper we continue our studies of the one dimensional conformal metric flows, which were introduced in math.AP/0611254. We prove the global existence and convergence of the one dimensional Yamabe and affine flows. Furthermore, we obtain exponential convergence of the metrics under these flows.

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arXiv2001

Global regularity of wave maps II. Small energy in two dimensions

We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.

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arXiv2011

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.

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