Nonlocal Conservation Laws in Bounded Domains

Colombo, Rinaldo M.; Rossi, Elena

doi:10.1137/18m1171783

Cited by 27 publications

(20 citation statements)

References 28 publications

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“…[23], Other sources, detailed in the proof below, are Refs. 14, 21, 22.Proposition Under the assumptions

(b)

(bold-italicc bold1)

, and

(q -)

, the Cauchy Problem () generates the map

\begin{matrix} right H & : & J & \times & U & \to & scriptU, \\ false(t_{o}, t false) & , & u_{o} & \to & u, \end{matrix}

where

u

is defined by (), with the following properties:

(boldH bold1)

H

is a process :

{scriptH}_{t, t} = Id

for all

t \in I

and

{scriptH}_{t_{2}, t_{3}} \circ {scriptH}_{t_{1}, t_{2}} = {scriptH}_{t_{1}, t_{3}}

for all

t_{1}, t_{2}, t_{3} \in I

, with

t_{1} \leq t_{2} \leq t_{3}

(boldH bold2)

Positivity : If

q \geq 0

and

u_{o} \in {scriptU}^{+}

, then

{scriptH}_{t_{o}, t} u_{o ...}

…”

Section: Analytic Proofsmentioning

confidence: 99%

Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system

Colombo

Rossi

2021

Stud Appl Math

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We develop a time and space‐dependent predator—prey model. The predators' equation is a nonlocal hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators “hunt” for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey–parasites. The general well‐posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilæ used to fight against the spreading of Drosophila suzukii.

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“…[23], Other sources, detailed in the proof below, are Refs. 14, 21, 22.Proposition Under the assumptions

(b)

(bold-italicc bold1)

, and

(q -)

, the Cauchy Problem () generates the map

\begin{matrix} right H & : & J & \times & U & \to & scriptU, \\ false(t_{o}, t false) & , & u_{o} & \to & u, \end{matrix}

where

u

is defined by (), with the following properties:

(boldH bold1)

H

is a process :

{scriptH}_{t, t} = Id

for all

t \in I

and

{scriptH}_{t_{2}, t_{3}} \circ {scriptH}_{t_{1}, t_{2}} = {scriptH}_{t_{1}, t_{3}}

for all

t_{1}, t_{2}, t_{3} \in I

, with

t_{1} \leq t_{2} \leq t_{3}

(boldH bold2)

Positivity : If

q \geq 0

and

u_{o} \in {scriptU}^{+}

, then

{scriptH}_{t_{o}, t} u_{o ...}

…”

Section: Analytic Proofsmentioning

confidence: 99%

Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system

Colombo

Rossi

2021

Stud Appl Math

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“…At the modeling level, the immigration of parasites is neglected in the present work. At the analytic level, general well posedness and stability results are currently apparently still missing, see [5] for recent preliminary results. The numerical algorithm to deal with boundary conditions would then be necessarily adapted.…”

Section: Optimized Timing Of Parasitoids' Releasesmentioning

confidence: 99%

On a Hyperbolic-Parabolic Parasitoid-Parasite System: Well Posedness and Control

Colombo,

Rossi

2020

Preprint

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We develop a time and space dependent predator -prey model. The predators' equation is a non local hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators "hunt" for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey-parasites. The general well posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations.The specific example we have in mind is that of Trichopria drosophilae used to fight against the spreading of Drosophila suzukii.

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“…Introduction. Moving crowds are usually modeled, at the macroscopic level, by evolution PDEs with nonlocal terms [5,9,7,8,17,16,19]. States of these equations are measures (or densities) which describe the distribution of individuals (also called agents) on some configuration space, typically, the space of agents' positions or position-velocity pairs.…”

mentioning

confidence: 99%

“…Stationary obstacles in both cases are handled by imposing either the Neumann ρv •n = 0 or the Dirichlet ρ = 0 boundary condition. The latter condition is more demanding: to achieve it one has to adjust nonlocal terms [8] or introduce specific distances in the space of measures [13].…”

mentioning

confidence: 99%

Modeling of crowds in regions with moving obstacles

Maltugueva¹,

Pogodaev²

2021

DCDS

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<p style='text-indent:20px;'>We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to <inline-formula><tex-math id="M1">\begin{document}$ r $\end{document}</tex-math></inline-formula>-prox-regular sets. The crowd motion obeys a nonlinear transport equation outside the obstacle and a normal cone condition (similar to that of the classical sweeping processes theory) on the boundary. We prove the well-posedness of the model, give an application to environment optimization problems, and provide some results of numerical computations.</p>

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Nonlocal Conservation Laws in Bounded Domains

Cited by 27 publications

References 28 publications

Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system

Well‐posedness and control in a hyperbolic–parabolic parasitoid–parasite system

On a Hyperbolic-Parabolic Parasitoid-Parasite System: Well Posedness and Control

Modeling of crowds in regions with moving obstacles

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