Maximal regularity and applications to PDEs

Emmrich, Etienne; Wittbold, Petra

doi:10.1515/9783110212105.247

Cited by 7 publications

(3 citation statements)

References 27 publications

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“…2 We refer to [24] for a survey on the subject. The very same estimate that we need is also presented, with a reference to the previous survey, in [8], formula (3), in the setting of Dirichlet conditions on a bounded domain.…”

Section: Proof Of Theorem B2 Step 1 Existencementioning

confidence: 99%

Second order local minimal-time Mean Field Games

Ducasse,

Mazanti,

Santambrogio

2020

Preprint

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The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain Ω in minimal expected time. The important point is that agents are constrained by a bound on the drift depending on the density of the other agents at their location (the higher the density, the smaller the velocity). Existence for a finite time horizon T is proven via a fixed point argument but, because of the diffusion, the model should be studied in infinite horizon as the total mass inside the domain decreases in time, but never reaches zero in finite time. Hence, estimates are needed to pass the solution to the limit as T → ∞, and the asymptotic behavior of the solution which is obtained in this way is also studied. This passes through a combination of classical parabolic arguments together with specific computations for MFGs. Both the Fokker-Planck equation ruling the evolution of the density of agents and the Hamilton-Jacobi-Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach ∂Ω. The initial datum for the density is given (and its regularity is discussed so as to have the sharpest result), and the long-time limit of the value function is characterized as the solution of a stationary problem. Contents 14 4.2. Improved convergence results 19 4.3. Initial conditions with finite entropy 21 Appendix A. Regularizing effects of parabolic equations 22 Appendix B. Fokker-Planck equation with finite-entropy initial condition 26 References 30

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Section: Proof Of Theorem B2 Step 1 Existencementioning

confidence: 99%

Second order local minimal-time Mean Field Games

Ducasse,

Mazanti,

Santambrogio

2020

Preprint

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“…Besides, according to the work by Lamberton in [13] the case p = r entails (3) in the general case. The reader may refer to the book [11] by Krylov or to the survey paper [19] by Monniaux for a 'modern' proof of (3) in the case µ = 0.…”

Section: Introductionmentioning

confidence: 99%

On the solvability of the compressible Navier–Stokes system in bounded domains

Danchin¹

2010

Nonlinearity

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This paper is dedicated to the well-posedness issue for the barotropic Navier-Stokes system with homogeneous Dirichlet boundary conditions in bounded domains of R N. We aim at considering data in as large a class as possible. Our main result is that if the initial density is bounded away from zero and belongs to some W 1,r with r > N, if the initial velocity is in the Besov space B 2−(2/p) r,p (and satisfies a suitable boundary condition), and if the body force is in L p loc (R + ; L r) for some p > 1 then the system has a unique local solution. Our regularity assumptions are consistent with a dimensional analysis which shows that critical data would correspond to r = N and p = 1, and improve an old result by Solonnikov (1980

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“…By (2.12) and (2.13) we have that Then, according to [36,Theorem 3.3], ∆ µ has the maximal L -regularity property in L q ((0, ∞)), that is, there exists C > 0 such that (6.82) R µ f L p ((0,∞),L q ((0,∞))) ≤ C f L p ((0,∞),L q ((0,∞))) ,…”

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confidence: 99%

Parabolic Equations Involving Bessel Operators and Singular Integrals

León-Contreras

2018

Integr. Equ. Oper. Theory

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In this paper we consider the evolution equation ∂tu = ∆µu + f and the corresponding Cauchy problem, where ∆µ represents the Bessel operator ∂ 2x + ( 1 4 − µ 2 )x −2 , for every µ > −1. We establish weighted and mixed weighted Sobolev type inequalities for solutions of Bessel parabolic equations. We use singular integrals techniques in a parabolic setting. Recently, Jones' results have been extended to weighted and mixed weighted L p spaces by Ping, Stinga and Torrea ([42, Theorem 2.4]). In [42], they also considered the equation (1.1) on the whole space R n+1 . In [42, Theorem 2.3], it was proved that if f is a C 2 -function defined on R n+1 with compact support, when n ≥ 2, the function u given byis a solution of (1.1) on R n+1 , where ∂ 2 xi,xj u(t, x) = lim ǫ→0 + Ωǫ

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Maximal regularity and applications to PDEs

Cited by 7 publications

References 27 publications

Second order local minimal-time Mean Field Games

Second order local minimal-time Mean Field Games

On the solvability of the compressible Navier–Stokes system in bounded domains

Parabolic Equations Involving Bessel Operators and Singular Integrals

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