A Partial Differential Equation Arising in a 1D Model for the 3D Vorticity Equation

Gregorio, S. De

doi:10.1002/(sici)1099-1476(199610)19:15<1233::aid-mma828>3.0.co;2-w

Cited by 53 publications

(39 citation statements)

References 6 publications

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“…• Similar results could be derived for other one-dimensional equations with non-local transport velocity [CCF05, DG90,DG96]. For instance, the so called CCF model [CCF05] is also known to produce singularities in finite time, although through a different mechanism to that of Burgers'.…”

Section: Discussionsupporting

confidence: 61%

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

Alonso-Orán

León

Takao

2019

Nonlinear Differ. Equ. Appl.

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In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

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Section: Discussionsupporting

confidence: 61%

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

Alonso-Orán

León

Takao

2019

Nonlinear Differ. Equ. Appl.

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“…For our purpose here we can think of the case when L is linear. The trajectory ξ(t) with ξ(t 0 ) = ξ 0 lies in the adjoint orbit 11) and hence the orbit invariants, such as eigenvalues of (a suitable representation of) ξ are preserved. The De Gregorio equation is formally of this form, with G = Diff + (S 1 ), the group of the orientation-preserving diffeomorphisms of the circle, except that Diff + (S 1 ) is infinitedimensional, and hence some of the properties of finite-dimensional Lie groups may not be available.…”

Section: Orbit Invariantsmentioning

confidence: 99%

“…where z is considered as a complex number and θ ∈ (−π, π]. De Gregorio [11] suggested the following modification of (1.1):…”

Section: Introductionmentioning

confidence: 99%

On the De Gregorio Modification of the Constantin–Lax–Majda Model

Jia

Stewart

Šverák

2018

Arch Rational Mech Anal

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We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM) model ωt = ω Hω on the unit circle. The De Gregorio equation is ωt + uωx − uxω = 0, ux = Hω. In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when t → ±∞, in a way resembling inviscid damping. We prove that such a behavior takes place near a manifold of equilibria.

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“…Numerical simulations of De Gregorio model [6] seem to suggest that there is no blow-up from smooth initial data on S 1 . Moreover, as mentioned by authors in [11], their numerical experiments seem to suggest that smooth solutions even converge to equilibria of (1.1)(and usually to the ground states A sin(θ + θ 0 )).…”

Section: Introductionmentioning

confidence: 97%

“…De Gregorio [6] suggested to include a convective term to the CLM model. The resulting modified equation reads ∂ t ω + u∂ θ ω = ω∂ θ u, ∂ θ u = Hω.…”

Section: Introductionmentioning

confidence: 99%

On the Constantin–Lax–Majda Model with Convection

Lei

Liu

Ren

2019

Commun. Math. Phys.

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The well-known Constantin-Lax-Majda (CLM) equation, an important toy model of the 3D Euler equations without convection, can develop finite time singularities [5]. De Gregorio modified the CLM model by adding a convective term [6], which is known important for fluid dynamics [10,14]. Presented are two results on the De Gregorio model. The first one is the global well-posedness of such a model for general initial data with non-negative (or non-positive) vorticity which is based on a newly discovered conserved quantity. This verifies the numerical observations for such class of initial data. The second one is an exponential stability result of ground states, which is similar to the recent significant work of Jia, Steward and Sverak [11], with the zero mean constraint on the initial data being removable. The novelty of the method is the introduction of the new solution space H DW together with a new basis and an effective inner product of H DW . Remark 1.4. Theorem 1.1 also holds true for non-positive initial data ω in , in which case we require that ω in ∈ L 1 (Ω) and ∂ θ ( √ −ω in ) ∈ L 2 (Ω). In fact, we only need to considerω = −ω, which satisfies: − ∂ tω +ū∂ θω =ω∂ θū , ∂ θū = Hω.(1.2)Note that equation (1.2) is simply a time reversed version of (1.1) and the proof of Theorem 1.1 in Section 2 still works.

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A Partial Differential Equation Arising in a 1D Model for the 3D Vorticity Equation

Cited by 53 publications

References 6 publications

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

On the De Gregorio Modification of the Constantin–Lax–Majda Model

On the Constantin–Lax–Majda Model with Convection

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