The energy estimate and uniqueness for linear parabolic equations #
Let V ↪ H ↪ V* be a Gelfand triple, given by a continuous linear map ι : V →L[ℝ] H from a
Banach space into a real inner product space (ContinuousLinearMap.gelfandDual). An abstract
linear parabolic equation on a time interval (a, b) is
u' + A(t) u = f,
where A(t) : V →L[ℝ] V* is a family of bounded operators, f ∈ L²(a, b; V*), and a weak
solution is a u ∈ L²(a, b; V) whose image ι.gelfandDual ∘ ι ∘ u in V* has a weak time
derivative u' ∈ L²(a, b; V*) satisfying the equation for almost every t. By
TauCeti.HasWeakLineDerivOn.exists_continuousOn_ae_eq_of_gelfandDual, ι ∘ u then has a
representative w continuous on [a, b], which gives meaning to the initial value w a.
For L = -∂ᵢ(aⁱʲ(x, t) ∂ⱼ ·) + … on a domain Ω, the triple is H¹₀(Ω) ↪ L²(Ω) ↪ H⁻¹(Ω) and
A(t) is the energy form of L at time t.
Assume that A(t) satisfies Gårding's inequality α ‖v‖² - β ‖ι v‖² ≤ A(t) v v for some
α > 0 and β ≥ 0. The energy identity d/dt ‖w‖² = 2 u'(u) = 2 f(u) - 2 A(t) u u, Young's
inequality 2 f(u) ≤ α⁻¹ ‖f‖² + α ‖u‖² and Grönwall's inequality give the energy estimate
‖w t‖² + α ∫ₐᵗ ‖u‖² ≤ exp (2 β (t - a)) (‖w a‖² + α⁻¹ ∫ₐᵗ ‖f‖²)
for every t ∈ [a, b] (TauCeti.HasWeakLineDerivOn.norm_sq_add_mul_integral_le_of_gelfandDual).
Applied to the difference of two solutions, it shows that a weak solution is determined by its
initial value and the right-hand side: their continuous representatives agree on [a, b]
(TauCeti.HasWeakLineDerivOn.eqOn_of_gelfandDual), and the solutions themselves agree almost
everywhere on (a, b) (TauCeti.HasWeakLineDerivOn.ae_eq_of_gelfandDual), even when ι is not
injective.
No measurability of t ↦ A(t) is assumed: the equation and the integrability of u'(u) and
f(u) already make t ↦ A(t) u(t) u(t) integrable.
Main declarations #
TauCeti.HasWeakLineDerivOn.norm_sq_add_mul_integral_le_of_gelfandDual: the energy estimate.TauCeti.HasWeakLineDerivOn.eqOn_of_gelfandDual: two weak solutions with the same initial value have the same continuous representative.TauCeti.HasWeakLineDerivOn.ae_eq_of_gelfandDual: two weak solutions with the same initial value agree almost everywhere.
References #
- L. C. Evans, Partial Differential Equations, 2nd ed., §7.1.2, Theorem 2 and Theorem 4.
- E. Zeidler, Nonlinear Functional Analysis and its Applications II/A, Chapter 23.
The energy estimate for linear parabolic equations. Let ι : V →L[ℝ] H define a Gelfand
triple, and let A(t) satisfy Gårding's inequality α ‖v‖² - β ‖ι v‖² ≤ A(t) v v with α > 0
and β ≥ 0 for almost every t ∈ (a, b). Let u ∈ L²(a, b; V) be such that
t ↦ ι.gelfandDual (ι (u t)) has weak derivative u' ∈ L²(a, b; V*) with u' + A(t) u = f almost
everywhere on (a, b), for some f ∈ L²(a, b; V*), and let w be continuous on [a, b] and equal
to ι ∘ u almost everywhere on (a, b). Then for every t ∈ [a, b],
‖w t‖² + α ∫ₐᵗ ‖u‖² ≤ exp (2 β (t - a)) (‖w a‖² + α⁻¹ ∫ₐᵗ ‖f‖²).
Uniqueness for linear parabolic equations: continuous representatives. Let ι : V →L[ℝ] H
define a Gelfand triple, and let A(t) satisfy Gårding's inequality
α ‖v‖² - β ‖ι v‖² ≤ A(t) v v with α > 0 and β ≥ 0 for almost every t ∈ (a, b). If u₁ and
u₂ are weak solutions in L²(a, b; V) of u' + A(t) u = f on (a, b), and their continuous
representatives w₁, w₂ on [a, b] have the same initial value w₁ a = w₂ a, then w₁ = w₂ on
[a, b].
Uniqueness for linear parabolic equations. Let ι : V →L[ℝ] H define a Gelfand triple, and
let A(t) satisfy Gårding's inequality α ‖v‖² - β ‖ι v‖² ≤ A(t) v v with α > 0 and β ≥ 0 for
almost every t ∈ (a, b). If u₁ and u₂ are weak solutions in L²(a, b; V) of
u' + A(t) u = f on (a, b), and their continuous representatives w₁, w₂ on [a, b] have the
same initial value w₁ a = w₂ a, then u₁ = u₂ almost everywhere on (a, b). This holds even when
ι is not injective.