Evolution equations governed by Lipschitz continuous non-autonomous forms

Ahmed Sani; Hafida Laasri

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 2, page 475-491
  • ISSN: 0011-4642

Abstract

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We prove L 2 -maximal regularity of the linear non-autonomous evolutionary Cauchy problem u ˙ ( t ) + A ( t ) u ( t ) = f ( t ) for a.e. t ∈ [ 0 , T ] , u ( 0 ) = u 0 , where the operator A ( t ) arises from a time depending sesquilinear form 𝔞 ( t , · , · ) on a Hilbert space H with constant domain V . We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of H .

How to cite

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Sani, Ahmed, and Laasri, Hafida. "Evolution equations governed by Lipschitz continuous non-autonomous forms." Czechoslovak Mathematical Journal 65.2 (2015): 475-491. <http://eudml.org/doc/270101>.

@article{Sani2015,
abstract = {We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy problem\[ \dot\{u\} (t)+A(t)u(t)=f(t) \quad \text\{for a.e.\ \} t\in [0,T],\quad u(0)=u\_0, \] where the operator $A(t)$ arises from a time depending sesquilinear form $\mathfrak \{a\}(t,\cdot ,\cdot )$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of $H.$},
author = {Sani, Ahmed, Laasri, Hafida},
journal = {Czechoslovak Mathematical Journal},
keywords = {sesquilinear form; non-autonomous evolution equation; maximal regularity; convex set},
language = {eng},
number = {2},
pages = {475-491},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Evolution equations governed by Lipschitz continuous non-autonomous forms},
url = {http://eudml.org/doc/270101},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Sani, Ahmed
AU - Laasri, Hafida
TI - Evolution equations governed by Lipschitz continuous non-autonomous forms
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 2
SP - 475
EP - 491
AB - We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy problem\[ \dot{u} (t)+A(t)u(t)=f(t) \quad \text{for a.e.\ } t\in [0,T],\quad u(0)=u_0, \] where the operator $A(t)$ arises from a time depending sesquilinear form $\mathfrak {a}(t,\cdot ,\cdot )$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of $H.$
LA - eng
KW - sesquilinear form; non-autonomous evolution equation; maximal regularity; convex set
UR - http://eudml.org/doc/270101
ER -

References

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