A variational approach to Navier–Stokes

Ortiz, Michael; Schmidt, Bernd; Stefanelli, Ulisse

doi:10.1088/1361-6544/aae722

Cited by 10 publications

(18 citation statements)

References 67 publications

(77 reference statements)

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“…and, arguing as in [15, proof of Theorem 2.5], one obtains convergence in H 1 ([0, T ]; L 2 ), (2) (with the latter meant as an equality in (W ∩ G) ′ ) and (16). On the other hand, (11), (12) and (14) imply…”

supporting

confidence: 58%

“…Step (iii): proof of (12)- (15). The proof of (12) and (13) is simply the repetition of the arguments developed in the first part of [21, Proof of Theorem 2.3: part (b)], with E ε (t) replaced by E d ε (t). Concerning (14), this is an immediate consequence of (60) in view of (62) and (10).…”

Section: Proof Of Theorem 25mentioning

confidence: 99%

“…[1,2,11], while for the applications to ODE systems we mention [7] (and references therein). We also quote some new extensions to the Navier-Stokes equation due to [12].…”

Section: Introductionmentioning

confidence: 99%

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An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

Tentarelli

Tilli

2019

Journal of Differential Equations

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We discuss a purely variational approach to the study of a wide class of second order nonhomogeneous dissipative hyperbolic PDEs. Precisely, we focus on the wave-like equations that present also a nonzero source term and a first-order-in-time linear term. The paper carries on the research program initiated in [14], and developed in [15,21], on the De Giorgi approach to hyperbolic equations.Hence, the link with (1) is immediate: as ε ↓ 0, supposing that f ε → f and w ε → w, one formally obtains (1) (and (2)) in the limit.A first comment is in order. The paper provides a purely variational method for proving existence of (weak) solutions to second order hyperbolic PDEs with dissipation, that is, wave-like

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supporting

confidence: 58%

Section: Proof Of Theorem 25mentioning

confidence: 99%

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An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

Tentarelli

Tilli

2019

Journal of Differential Equations

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“…This approach which consist in minimizing an appropriate norm of the solution is refereed to in the literature as variational approach. We mention notably the work [16] where strong solution of (1.1) are characterized in the two dimensional case in term of the critical points of a quadratic functional, close to E. Similarly, the authors in [15] show that the following functional…”

Section: Introductionmentioning

confidence: 94%

A Fully Space-Time Least-Squares Method for the Unsteady Navier–Stokes System

Lemoine

Münch

2021

J. Math. Fluid Mech.

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We introduce and analyze a space-time least-squares method associated with the unsteady Navier-Stokes system. Weak solution in the two dimensional case and regular solution in the three dimensional case are considered. From any initial guess, we construct a minimizing sequence for the least-squares functional which converges strongly to a solution of the Navier-Stokes system. After a finite number of iterations related to the value of the viscosity coefficient, the convergence is quadratic. Numerical experiments within the two dimensional case support our analysis. This globally convergent leastsquares approach is related to the damped Newton method.

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“…The WED variational approach has been applied to a variety of different parabolic problems, including gradient flows [16][17][18][19], rate-independent flows [20,21], crack propagation [22], doubly-nonlinear flows [23][24][25][26][27], nonpotential perturbations [28,29] and variational approximations [30], curves of maximal slope in metric spaces [31][32][33], mean curvature flow [13,34], dynamic plasticity [35], and the incompressible Navier-Stokes system [36].…”

Section: Introductionmentioning

confidence: 99%

Stochastic PDEs via convex minimization

Scarpa

Stefanelli

2020

Communications in Partial Differential Equations

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We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.

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A variational approach to Navier–Stokes

Cited by 10 publications

References 67 publications

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

An existence result for dissipative nonhomogeneous hyperbolic equations via a minimization approach

A Fully Space-Time Least-Squares Method for the Unsteady Navier–Stokes System

Stochastic PDEs via convex minimization

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