Superposition principle for the continuity equation in a bounded domain

Bonicatto, Paolo; Гусев, Н. А.

doi:10.1088/1742-6596/990/1/012002

Cited by 5 publications

(7 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It was first introduced in the Euclidean setting by Ambrosio in [4], where it was employed to investigate uniqueness and stability of Lagrangian flows in the context of DiPerna-Lions Theory [23]. Since then it has been applied to different tasks [5,10,11,12,13] and extended to various settings [15,31,34,41]. In [4] the velocity field v is assumed to satisfy (5) ˆ1 0 ˆΩ |v(t, x)| 2 dρ t (x) dt < ∞ .…”

Section: Introductionmentioning

confidence: 99%

A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

Bredies¹,

Carioni²,

Fanzon³

2020

Preprint

View full text Add to dashboard Cite

We study measure-valued solutions of the inhomogeneous continuity equation ∂ t ρ t + div(vρ t ) = gρ t where the coefficients v and g are of low regularity. A new superposition principle is proven for positive measure solutions and coefficients for which the recently-introduced dynamic Hellinger-Kantorovich energy is finite. This principle gives a decomposition of the solution into curves t → h(t)δ γ(t) that satisfy the characteristic system γ(t) = v(t, γ(t)), ḣ(t) = g(t, γ(t))h(t) in an appropriate sense. In particular, it provides a generalization of existing superposition principles to the low-regularity case of g where characteristics are not unique with respect to h. Two applications of this principle are presented. First, uniqueness of minimal total-variation solutions for the inhomogeneous continuity equation is obtained if characteristics are unique up to their possible vanishing time. Second, the extremal points of dynamic Hellinger-Kantorovichtype regularizers are characterized. Such regularizers arise, e.g., in the context of dynamic inverse problems and dynamic optimal transport.

show abstract

Section: Introductionmentioning

confidence: 99%

A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

Bredies¹,

Carioni²,

Fanzon³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Further, for each (a x,ȳ ) x,ȳ∈S ∈ A(β 1 S , β 2 S ), we have that Taking the minimum w.r.t. elements of A(β 1 , β 2 ), we arrive at the second inequality in (20).…”

Section: Appendix a Distances Between Measures On A Finite Setmentioning

confidence: 96%

Nonlocal balance equation: representation of solution and Markov approximation

Averboukh

2024

Preprint

View full text Add to dashboard Cite

We study the nonlocal balance equation that describes the evolution of a system consisting of infinitely many identical particles those move along a deterministic dynamics and can also either disappear or give a spring. In this case, the solution of the balance equation is considered in the space of nonnegative measures. We prove the superposition principle for the examined nonlocal balance equation. Furthermore, we interpret the source/sink term as a probability rate of jumps from/to a remote point. Using this idea and replacing the deterministic dynamics of each particle by a nonlinear Markov chain, we approximate the solution of the balance equation is approximated by a solution of a system of ODEs and evaluate the corresponding approximation rate. MSC Classification: 35R06, 70F45, 60J27

show abstract

“…Extend the particle trajectories from (1.9) by X x t := lim s↑tx X x s ∈ ∂Ω for all t ≥ t x , and let Ω t := {X x t | x ∈ Ω & t x > t} for all t ∈ [0, T ) (these sets are open due to (3.1)). Then the lemma essentially follows from Theorem 2 in [3] but in order to apply it, we need to show that ω weakly satisfies some boundary conditions on (0, T ) × ∂Ω (even though these do not affect the result). To this end we employ Theorem 3.1 and Remark 3.1 in [4], which show that there is indeed some κ ∈ L ∞ ((0, T ) × ∂Ω) such that…”

Section: Proof Of Theorem 11(i)mentioning

confidence: 99%

“…(In fact, the measure in [3] is supported on the set of all maximal solutions to the ODE d dt Y (t) = u(t, Y (t)) on (0, T ), and the relevant formula holds for all ψ ∈ C ∞ 0 (R d ). But this becomes (3.2) when restricted to the ψ above, with η the restriction of the measure from [3] to the set of solutions {{X x t } t∈(0,T ) | x ∈ Ω}. This is because uniqueness of solutions for the ODE shows that the other solutions have Y (t) /…”

Section: Proof Of Theorem 11(i)mentioning

confidence: 99%

See 1 more Smart Citation

Euler Equations on General Planar Domains

Han¹,

Zlatoš²

2020

Preprint

View full text Add to dashboard Cite

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. This condition is only slightly more restrictive than exclusion of corners with angles greater than π and, in particular, is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.

show abstract

Superposition principle for the continuity equation in a bounded domain

Cited by 5 publications

References 8 publications

A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

A superposition principle for the inhomogeneous continuity equation with Hellinger-Kantorovich-regular coefficients

Nonlocal balance equation: representation of solution and Markov approximation

Euler Equations on General Planar Domains

Contact Info

Product

Resources

About