L-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

Angiuli, Luciana; Lorenzi, Luca; Pallara, Diego

doi:10.1016/j.jmaa.2016.06.001

Cited by 20 publications

(33 citation statements)

References 18 publications

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“…For antisymmetric potentials , a linear growth of the entries is possible. In this article, we will allow potential terms whose entries grow like | | for some ∈ [1,2). Note that the results obtained here do not follow from [18] by setting = 0.…”

Section: Introductionmentioning

confidence: 95%

“…The strategy in these references is quite different from that in . Namely, in solutions to the parabolic equation are at first constructed in the space of bounded and continuous functions. Afterwards the semigroup is extrapolated to the

L^{p}

‐scale.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the results obtained here do not follow from [18] by setting = 0. Following [18], there were some other publications [1,2,11] with less restrictive assumptions on the coefficients; in particular, also unbounded diffusion coefficients were considered. The strategy in these references is quite different from that in [18].…”

Section: Introductionmentioning

confidence: 99%

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‐Theory for Schrödinger systems

Kunze

Lorenzi

Maichine

et al. 2019

Mathematische Nachrichten

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In this article, we study for p∈(1,∞) the Lp‐realization of the vector‐valued Schrödinger operator Lu:=prefix div (Q∇u)+Vu. Using a noncommutative version of the Dore–Venni theorem due to Monniaux and Prüss, we prove that the Lp‐realization of scriptL, defined on the intersection of the natural domains of the differential and multiplication operators which form scriptL, generates a strongly continuous contraction semigroup on Lp(double-struckRd;double-struckCm). We also study additional properties of the semigroup such as extension to L1, positivity, ultracontractivity and prove that the generator has compact resolvent.

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Section: Introductionmentioning

confidence: 95%

L^{p}

‐scale.…”

Section: Introductionmentioning

confidence: 99%

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‐Theory for Schrödinger systems

Kunze

Lorenzi

Maichine

et al. 2019

Mathematische Nachrichten

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“…Taking advantage of the results in [2], the authors of [6] provide sufficient conditions for the semigroup T (t) to admit a bounded extension to L p (R d ; R m ). Also some summability improving properties of the semigroup are studied.…”

Section: Introductionmentioning

confidence: 99%

On coupled systems of PDEs with unbounded coefficients

Angiuli

Lorenzi

2020

Dynamics of Partial Differential Equations

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We study the Cauchy problem associated to parabolic systems of the form Dtu = A(t)u in C b (R d ; R m ), the space of continuous and bounded functions f : R d → R m . Here A(t) is a weakly coupled elliptic operator acting on vector-valued functions, having diffusion and drift coefficients which change from equation to equation. We prove existence and uniqueness of the evolution operator G(t, s) which governs the problem in C b (R d ; R m ) proving its positivity. The compactness of G(t, s) in C b (R d ; R m ) and some of its consequences are also studied. Finally, we extend the evolution operator G(t, s) to the L pspaces related to the so called "evolution system of measures" and we provide conditions for the compactness of G(t, s) in this setting.2000 Mathematics Subject Classification. 35K40, 35K45, 37L40, 46B50, 47A15.

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“…We mention the article [12] where systems of parabolic equations coupled through both, a potential term and a drift term, were considered in the L p -setting. In [1,3,6], the authors choose a different approach. Indeed, they first constructed solutions in the space of bounded and continuous functions and only afterwards the obtained semigroup is extrapolated to the L p -scale.…”

Section: Introductionmentioning

confidence: 99%

Vector-valued Schrödinger operators in <i>L<sup>p</sup></i>-spaces

Kunze¹,

Maichine²,

Rhandi³

2018

Discrete &Amp; Continuous Dynamical Systems - S

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In this paper we consider vector-valued Schrödinger operators of the form div(Q∇u) − V u, where V = (v ij ) is a nonnegative locally bounded matrix-valued function and Q is a symmetric, strictly elliptic matrix whose entries are bounded and continuously differentiable with bounded derivatives. Concerning the potential V , we assume an that it is pointwise accretive and that its entries are in L ∞ loc (R d ). Under these assumptions, we prove that a realization of the vector-valued Schrödinger operator generates a C 0 -semigroup of contractions in L p (R d ; C m ). Further properties are also investigated.2010 Mathematics Subject Classification. Primary: 35K40, 47D08; Secondary: 47D06.

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L-estimates for parabolic systems with unbounded coefficients coupled at zero and first order

Cited by 20 publications

References 18 publications

‐Theory for Schrödinger systems

‐Theory for Schrödinger systems

On coupled systems of PDEs with unbounded coefficients

Vector-valued Schrödinger operators in <i>L<sup>p</sup></i>-spaces

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