|
|
| |
| Well-Posedness of Stochastic Continuity Equations onRiemannian Manifolds |
| |
Citation： |
Luca GALIMBERTI,Kenneth H. KARLSEN.Well-Posedness of Stochastic Continuity Equations onRiemannian Manifolds[J].Chinese Annals of Mathematics B,2024,45(1):81~122 |
| Page view： 342
Net amount： 441 |
Authors： |
Luca GALIMBERTI; Kenneth H. KARLSEN |
Foundation： |
This work was supported by the Research Council of Norway through the projects Stochastic Conservation Laws (No. 250674) and (in part) Waves and Nonlinear Phenomena (No. 250070). |
|
|
| Abstract： |
The authors analyze continuity equations with Stratonovich stochasticity,
?ρ + divh
h
ρ ?
u(t, x) +XN
i=1
ai(x)W˙
i(t)
i = 0
defined on a smooth closed Riemannian manifold M with metric h. The velocity field u is
perturbed by Gaussian noise terms W˙
1(t), · · · , W˙ N (t) driven by smooth spatially dependent
vector fields a1(x), · · · , aN (x) on M. The velocity u belongs to L
1
tW1,2
x with divh u bounded
in L
p
t,x for p > d + 2, where d is the dimension of M (they do not assume divh u ∈ L
∞t,x).
For carefully chosen noise vector fields ai (and the number N of them), they show that the
initial-value problem is well-posed in the class of weak L
2
solutions, although the problem
can be ill-posed in the deterministic case because of concentration effects. The proof of this
“regularization by noise” result is based on a L
2
estimate, which is obtained by a duality
method, and a weak compactness argument. |
Keywords： |
Stochastic continuity equation, Riemannian manifold, Hyperbolic equation, Non-smooth velocity field, Weak solution, Existence, Uniqueness |
Classification： |
60H15, 35L02, 58J45, 35D30 |
|
|
Download PDF Full-Text
|
|
|
|
