Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic &amp; 2D-Navier–Stokes Equations: The General Case

Debbi, Latifa

doi:10.1007/s00021-020-00509-3

Cited by 4 publications

(6 citation statements)

References 74 publications

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“…Furthermore, we do not require that v is divergence-free, which means that our result applies to compressible fluids as well. Such a result is new, as far as we know, since v is different from those considered in other works such as [45] and [19]. However, by (55), v must be bounded, which is a severe restriction, and therefore, in general, it may not cover the subcritical quasi-geostrophic model where v is given by the vorticity function of the Riesz transform of u.…”

Section: Anisotropic Advection-diffusion Fractional Equations For S > 1mentioning

confidence: 83%

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

K.¹,

Rodrigues²

2022

Preprint

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Consider the quasilinear diffusion problem] m and any T ∈]0, ∞[, where Σu ∈ R q for 0 < q ≤ m × n represents fractional or nonlocal derivatives with order σ with σ < 2s for all 0 < s ≤ 1, including the classical gradient and derivatives of order greater than 1. We show global existence results for various quasilinear diffusion systems in non-divergence form, for different linear operators A, including local elliptic systems, anisotropic fractional equations and systems, and anisotropic nonlocal operators, of the following type (Au) i = − α,β,j ∂α(A αβ ij ∂ β u j ), Au = −D s (A(x)D s u), and (Au) i = ˆRn Aij(x, y) u j (x) − u j (y) |x − y| n+2s dy, for coercive, invertible matrices Π and suitable vectorial functions f .

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Section: Anisotropic Advection-diffusion Fractional Equations For S > 1mentioning

confidence: 83%

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

K.¹,

Rodrigues²

2022

Preprint

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“…Furthermore, we recall, see, for example, Debbi 30 and Taylor 42, ps. 283, 303 that A : D ( A ) → L 2 (0,1) is an isomorphism, the inverse A −1 is self adjoint, and thanks to the compact embedding of D ( A ) in L 2 (0,1), we conclude that A −1 is compact on L 2 (0,1).…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…The process

scriptW

has been studied, for example, in previous studies, 23,44,46 where the authors proved that for α ∈ (1,2], the Ornstein–Uhlenbeck process () is well defined as an L 2 ‐valued stochastic process with

C_{t}^{\frac{α - 1}{2 α}} C_{x}^{\frac{α - 1}{2}} -

Hölder continuous trajectories. In Debbi, 30 the author proved that for α ∈ (1,2],

scriptW \in L^{p} false(normalΩ; C_{t} \times H_{q}^{β} false) false(c false(0, 1 false) false)

, 2 ≤ q < ∞ , β ≥ 0, p ≥ 2, and c (0,1) is the unit circle. The following Lemma is a generalization of Blömker and Jentzen 9, Proposition 4.2.…”

Section: Some Estimates For the Stochastic Termsmentioning

confidence: 99%

“…The author established thresholds guaranteeing the wellposedness of these equations, according to the kind of solutions (strong, weak, martingale) and according to the Sobolev and the integrability regularities required. A generalization of this class of equations on

D \subseteq ℝ^{d}

and their wellposedness have also been studied in Debbi et al 31 …”

Section: Introductionmentioning

confidence: 98%

“…To the best knowledge of the authors, Göyngy was the first to introduce stochastic Burgers‐type equation, 3 then several works followed, see, for example, previous studies 1,4,29 . Other kind of generalization has been investigated by the second author in Debbi 30 . In this work, the author introduced a new class of multidimensional stochastic active scalar equations covering among others the multidimensional quasi‐geostrophic, 2 D ‐Navier–Stokes and the fractional stochastic Burgers equation on the torus with α ∈ (1,2].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

Arab

Debbi

2020

Math Methods in App Sciences

Self Cite

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In this work, we use the spectral Galerkin method to prove the existence of a pathwise unique mild solution of a fractional stochastic partial differential equation of Burgers type in a Hölder space. We get the temporal regularity, and using a combination of Galerkin and exponential-Euler methods, we obtain a full discretization scheme of the solution. Moreover, we calculate the rates of convergence for both approximations (Galerkin and full discretization) with respect to time and to space.

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Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

Lo,

Rodrigues

2024

Mathematische Nachrichten

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We consider the quasilinear diffusion problem of for an open set , , for , and any . Here, denotes an operator which may involve the distributional Riesz fractional gradient of order , with , the classical gradient or/and nonlocal derivatives , with . We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators , including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: for coercive, invertible matrices and suitable vectorial functions .

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Fractional Stochastic Active Scalar Equations Generalizing the Multi-Dimensional Quasi-Geostrophic & 2D-Navier–Stokes Equations: The General Case

Cited by 4 publications

References 74 publications

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

Global Existence for Nonlocal Quasilinear Diffusion Systems in Non-Isotropic Non-Divergence Form

Fractional stochastic Burgers‐type equation in Hölder space—Wellposedness and approximations

Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

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