On the maximal L-L regularity of solutions to a general linear parabolic system (original) (raw)

On the maximal L_p-L_q regularity of solutions to a general linear parabolic system

2019

We show the existence of solution in the maximal L_p-L_q regularity framework to a class of symmetric parabolic problems on a uniformly C^2 domain in R^n. Our approach consist in showing R - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition.

Regularity of a Parabolic Equation Solution in a Nonsmooth and Unbounded Domain

Journal of the Australian Mathematical Society, 2008

This work is concerned with the problem ∂ t u − c(t)∂ 2 x u = f u |∂ D\ T = 0 posed in the domain D = {(t, x) ∈ R 2 | 0 < t < T, ϕ 1 (t) < x < ϕ 2 (t)}, which is not necessary rectangular, and with T = {(T, x) | ϕ 1 (T) < x < ϕ 2 (T)}. Our goal is to find some conditions on the coefficient c and the functions (ϕ i) i=1,2 such that the solution of this problem belongs to the Sobolev space H 1,2 (D) = {u ∈ L 2 (D) | ∂ t u ∈ L 2 (D), ∂ x u ∈ L 2 (D), ∂ 2 x u ∈ L 2 (D)}.

A regularity theorem for parabolic equations

Journal of Functional Analysis, 1971

We consider the solution in a Hilbert space H of a parabolic equation of the following type: u'(t) + A(t) u(t) = 0; 40) = %I I where A(t) is an elliptic operator depending on t. We prove, under suitable hypotheses on A(t), an abstract regularity theorem, generalising the usual result (see J. L. LIONS, "Equations DifErentielles

An Lp-approach for the study of degenerate parabolic equations

Electronic Journal of Differential Equations

We give regularity results for solutions of a parabolic equation in non-rectangular domains U = ∪ t∈]0,1[ {t} × It with It = {x : 0 < x < ϕ(t)}. The optimal regularity is obtained in the framework of the space L p with p > 3/2 by considering the following cases: (1) When ϕ(t) = t α , α > 1/2 with a regular right-hand side belonging to a subspace of L p (U) and under assumption p > 1 + α. We use Labbas-Terreni results [11]. (2) When ϕ(t) = t 1/2 with a right-hand side taken only in L p (U). Our approach make use of the celebrated Dore-Venni results [2].

Maximal regularity for parabolic equations with measurable dependence on time and applications

2013

The subject of this thesis is the study of maximal Lp-regularity of the Cauchy problem u'(t)+A(t)u(t)=f(t), t∈ (0,T), u(0)=x. We assume (A(t))_{t∈ (0,T)} to be a family of closed operators on a Banach space X0, with constant domain D(A(t))=X1 for every t∈ (0,T). Maximal Lp-regularity means that for all f∈ Lp(0,T;X0), the solution of the above evolution problem is such that u', Au are both in Lp(0,T;X0). In the first part of the thesis, we introduce a new operator theoretic approach to maximal Lp-regularity in the case the dependence t→A(t) is just measurable. The abstract method is then applied to concrete parabolic PDEs: we consider equations and systems of elliptic differential operators of even order, with coefficients measurable in the time variable and continuous in the space variables, and we show that they have maximal Lp-regularity on Lq(\Rd), for every p,q∈(1,∞). These results gives an alternative approach to several PDE results in the literature, where only the ...

On a parabolic equation in a triangular domain

Applied Mathematics and Computation, 2002

We prove the optimal regularity, in Sobolev spaces, of the solution of a parabolic equation set in a triangular domain T. The right-hand term of the equation is taken in Lebesgue space L p ðT Þ. The method of operators sums in the non-commutative case is referred to.