Part IX, Chapter 52 Parabolic PDEs: theory and space approximation We introduce in this chapter some basic properties of parabolic equations and investigate various approximation techniques. The reader is strongly encour- aged to consult Thom´ee [478] for a thorough analysis of parabolic equations. 52.1 Mathematical analysis 52.1.1 Functional setting and Bochner integral In this section we review some basic concepts of functional analysis which are useful in dealing with time-dependent functions with values in a Ba- nach space, see Kufner et al. [332, 2.19]. Let I be an nonempty bounded open set in R. Let V be a Banach§ space (real or complex). We say that f : I V is a simple function if there exit a collection vk k 1: m −→ { } ∈{ } of elements in V , m N, and disjoint measurable subsets Ak k 1: m ∈ 1 { } ∈{ } in I such that f(t) = k 1: m vk Ak (t) for all t I. The Bochner in- ∈{ } ∈ tegral of a simple function is defined by I f(t) dt := k 1: m vk Ak . P ∈{ } | | Clearly f(t) dt f(t) dt for any simple function f. We say say I V I V R P f : I k V is stronglyk ≤ measurablek k if there is a countable sequence of simple −→ R R functions fn n N such that limk f(t) fk(t) V = 0 for a.e. t in I. { } ∈ →∞ k − k Lemma 52.1. Let f : I V and assume that f is strongly measurable, −→ then the mapping I t f(t) V R is Lebesgue measurable. ∋ 7−→ k k ∈ Proof. See [332, Lem. 2.19.2]. ⊓⊔ Definition 52.2. We say that f : I V is Bochner integrable if there ex- −→ ists a countable sequence of simple functions fn n N such that limk f(t) { } ∈ →∞ k − fk(t) V = 0 a.e. t in I, and limk I f(t) fk(t) V dt = 0. Let f be Bochnerk integrable function and B →∞I bek a measurable− k set; the Bochner in- ⊂ R tegral of f over B is defined by f(t) dt := limk 1B(t)fk(t) dt. B →∞ I R R 726 Chapter 52. Parabolic PDEs: theory and space approximation The coherence of this definition is verified in Exercise 52.1. Theorem 52.3 (Bochner). A strongly mesurable function f : I V is −→ Bochner integrable if and only if f(t) V dt< . I k k ∞ Proof. See [332, Thm. 2.19.8] R ⊓⊔ In this book we are only going to manipulate strongly measurable functions. Then the above theorem says that to verify that f is Bochner integrable it 1 suffices to verify that I t f(t) V R is in L (I; R). ∋ 7−→ k k ∈ Definition 52.4. For 1 p + , we define Lp(I; V ) to be the space of V -valued functions that are≤ strongly≤ ∞ measurable and such that the following norm is finite: 1 p p I u(t) V dt if 1 p< + , u Lp(I;V ) = k k ≤ ∞ (52.1) k k (ess supt I u(t) V if p =+ . R ∈ k k ∞ ( It follows from the above definitions and results that Lp(I; V ) ֒ L1(I; V →p and I f(t) dt V I f(t) V dt := f L1(I;V ) for every f L (I; V ). We henceforthk denotek by≤ C0k(I; V )k the spacek composedk of the functions∈ u : I V R R −→ such that the mapping I t u(t)V V is continuous. We also denote 0 0 ∋ −→ k k C (I; V )= C (I; V ) L∞(I; V ). These two spaces are Banach spaces. ∩ Theorem 52.5. The space Lp(I; V ) is a Banach space for all p [1, ]. ∈ ∞ Proof. See [332, Thm. 2.20.4] ⊓⊔ Definition 52.6. Given q [1, ] and a second Banach space W with contin- ( uous embedding V ֒ W , we∈ say∞ that u has a weak derivative in Lq((0,T ); W if there is w Lq((0→,T ); W ) such that ∈ T T φ′(t)u(t) dt = φ(t)w(t) dt, φ C∞((0,T ); R), (52.2) − ∀ ∈ 0 Z0 Z0 and we denote ∂tu := w. Henceforth we set I := (0,T ) with T > 0. Since time-evolution problems are initial-value problems, it is important to determine which type of functions have a trace over the time interval (0,T ). The following Lemma gives an answer to this question. Lemma 52.7. Let 1 p, q + , let B0 ֒ B1 be two Banach spaces with continuous embeddings,≤ and≤ set ∞ → p q (B ,B ) := v L ((0,T ); B ) ∂tv L ((0,T ); B ) . W 0 1 { ∈ 0 | ∈ 1 } (i) (B0,B1) is a Banach space when equipped with the norm u (B ,B ) = W k kW 0 1 u p + ∂tu q . k kL ((0,T );B0) k kL ((0,T );B1) Part IX. Time-Dependent PDEs 727 (ii) (B ,B ) is continuously embedded in C0([0,T ]; B ). W 0 1 1 Proof. See Exercise 52.4 for the proof of item (ii). ⊓⊔ We henceforth denote u C0([0,T ];B) = supt [0,T ] u(t) B and u C1([0,T ];B) = k k ∈ k k | | supt [0,T ] ∂tu(t) B, etc. ∈ k k ֒ Lemma 52.8 (Aubin-Lions-Simon). Let 1 p, q + and let B0 B ֒ B be three Banach spaces with B compactly≤ embedded≤ ∞ in B and →B → 1 0 continuously embedded in B1. p .i) The embedding (B0,B1) ֒ L ([0,T ]; B) is compact) W → 0 ii) The embedding (B0,B1) ֒ C ([0,T ]; B) is compact if p = and) q > 1. W → ∞ Proof. See Aubin [23]; see also, e.g., Amann [9], Lions [353], Lions and Ma- genes [354], Simon [452]. ⊓⊔ Henceforth, we specialize the above setting by restricting ourselves to Hilbert spaces and taking p = q = 2 in Lemma 52.7. Let V ֒ L be two Hilbert spaces with continuous embedding. The norm of the embeddin→ g op- 1 erator is denoted by cP− , i.e., v V, cP v L v V . (52.3) ∀ ∈ k k ≤k k We assume that V is dense in L, and we identify L with L′ so that we are in the situation where V ֒ L L′ ֒ V ′, i.e., the duality paring , V ′,V can be → ≡ → h· ·i viewed as an extension of the inner product in L. Note that cP f V ′ f L k k ≤k k and f, v V ′,V = (f, v)L for all f L and all v V . In this setting, the followingh i result justifies integration by∈ parts with respect∈ to time (see Dautray and Lions [190, Thm. 2, p. 477]): Lemma 52.9 (Integration by parts). Under the above assumptions, for all u, v (V, V ′), the following identity holds: ∈W T T ∂tu(t), v(t) V ′,V dt = (u(T ), v(T ))L (u(0), v(0))L ∂tv(t),u(t) V ′,V dt. h i − − h i Z0 Z0 52.1.2 Well-posedness We now state a general result for time-dependent problems which plays a similar role to that played by the Lax–Milgram Lemma for elliptic equations. Let V ֒ L L′ ֒ V ′ be a Hilbertian setting as defined above. Consider a mapping→a :≡ (0,T )→V V R such that a(t, , ) is bilinear for a.e. t in (0,T ). Moreover, assume× × that→a satisfies the following· · properties: (p1) The function t a(t,u,v) is measurable u, v V . 7→ ∀ ∈ (p2) M such that a(t,u,v) M u V v V for a.e. t [0,T ], u, v V . ∃ | |≤ k k k k ∈ ∀ ∈ 728 Chapter 52. Parabolic PDEs: theory and space approximation 2 2 (p3) α> 0 and γ > 0 such that a(t,u,u) α u V γ u L for a.e. t [0,T ] and∃ for all u V . ≥ k k − k k ∈ ∈ 2 For f L ((0,T ); V ′) and u L, consider the following problem: ∈ 0 ∈ Find u (V, V ′) such that ∈W ∂tu, v V ′,V + a(t,u,v)= f(t), v V ′,V , a.e. t (0,T ), v V, (52.4) h i h i ∈ ∀ ∈ u(0) = u0. The initial data u(0) = u0 is meaningful according to Lemma 52.8. Note that the time evolution equation has to be understood in the distribution sense, T T i.e., ( φ′(t)(u(t), v)L + φ(t)a(t,u(t), v)) dt = φ(t) f(t), v V ′,V dt, for 0 − 0 h i all φ ∞((0,T ); R). R∈C0 R Definition 52.10 (Parabolic equation). Equation (52.4) is said to be parabolic whenever the bilinear form a satisfies the conditions (p1), (p2), and (p3). Up to a change of variable, it is always possible to modify condition (p3) so that γ = 0. Indeed, upon setting a˜(t, φ, v)= a(t, φ, v)+ γ(φ, v)L, it is clear thata ˜ satisfies conditions (p1) and (p2), and thata ˜ is V -coercive 2 γt γt sincea ˜(t, φ, φ) α φ V . Furthermore, setting φ = e− u and g = e− f, problem (52.4) is≥ recastk k in the following equivalent form: Find φ (V, V ′) such that ∈W ∂tφ, v V ′,V +˜a(t, φ, v)= g(t), v V ′,V , a.e. t (0,T ), v V, (52.5) h i h i ∈ ∀ ∈ φ(0) = u0.