Outline Velocity averaging a general framework Martin Lazar BCAM ERC-NUMERIWAVES Seminar May 15, 2013 Joint work with D. Mitrović, University of Montenegro, Montenegro
Outline Outline 1 2 L p, p >= 2 setting 3 with discontinuous coefficients 4 L p,
P a differential operator of the form Pu(x, y) = ( Aα (x, y)u(x, y) ), α k α x x R d a physical/space variable, y R m - a velocity variable. We consider a sequence of problems Pu n = g n, where (g n ) converges strongly in some suitable Sobolev space. Can we get compactness properties of (u n ) in L p? No! E.g. u n (x, y) = sin(ny), with x-independent coefficients. However (in applications) it is enough to analyse the averaged sequence u n (x, y)ρ(y)dy, ρ C c (R m y ). Velocity averaging lemma: to state conditions on (u n ), P and (Pu n ) such that a sequence of averaged quantities is relatively compact in L p loc.
Velocity averaging The first results: for the first order operators with constant coefficients, usually obtained by an application of the Fourier transform. Heterogeneous velocity averaging problem application of (a variant of) defect measure. Standard defect measures do not distinguish oscillations of different frequencies. In order to control oscillations, a microlocal variant is proposed (H-measures!).
H-measures - introduced around 1990 by L. Tartar and P. Gérard - describe quadratic limit behaviour of sequences converging weakly in L 2 - positive Radon measures on R d S d 1 (S d 1 unit sphere in R d ). If u n 0 in L 2 (R d ) then û n 0 in L 2 loc (Rd ). In the lack of strong convergence some information goes toward in the dual space. The dual space is projected onto S d 1 (by ξ/ ξ ). ξ 2... ξ d T T 0 1 ξ 1
H-measures Theorem 1. (Existence of H-measures) Let u n 0 in L 2 (R d ). There exists a subsequence (u n ) and a non-negative Radon measure µ on R d S d 1 such that for all ϕ 1, ϕ 2 C 0 (R d ), ψ C(S d 1 ): lim n R d F ( ϕ 1 u n ) F ( ϕ 2 u n ) ψ ( ) ξ dξ = µ, (ϕ 1 ϕ 2 )ψ ξ = ϕ 1 (x) ϕ 2 (x)ψ(ξ) dµ(x, ξ). R d S d 1 The theorem is also valid: for u n of class L 2 loc, but the associated H-measure does not need to be finite, test functions ϕ C c (R d ), for vector functions u n L 2 (R d ; C r ), the H-measure is a positive semi-definite matrix Radon measure.
H-measures vs. defect measures (u n ) bounded in L 2 = (u 2 n) bounded in L 1 Connection with defect measure: Corollary 1. u n L 2 (R d ) such that u n u n converges weakly (vaguely) to measure ν, then for every ϕ C 0 (R d ) ν, ϕ = µ, ϕ(x) 1(ξ), where µ is an H-measure associated to (sub)sequence (u n ). In contrast to defect measure, H-measure distinguishes oscillations corresponding to different frequencies (dual variable ξ) - microlocal defect measures. Simple example: u n (t, x) = sin (2π(αnt + βnx)) The associated H-measure µ = 1 ( δ(α0,β 4 0)(τ, ξ) + δ ( α0, β 0)(τ, ξ) ) α λ(t, x), α 0 = α 2 + β 2, β β 0 = α 2 + β 2.
Trivial H-measures To a sequence converging strongly, it corresponds the H-measure zero. The opposite is true, but only in L 2 loc. The notion also extends to functions taking values in an infinite dimensional Hilbert space H (instead of C r ). However, in that case µ = 0 does not imply strong convergence of (a sub) sequence (of) (u n ). Corollary 2. For a sequence (u n ) L 2 loc (Rd ; H) the associated H-measure µ equals zero iff the sequence ( u n h ) is relatively compact in L 2 loc (Rd ) for every h H ( relative compactness modulo H).
A classical example Let (u n ) a bounded sequence in L 2 loc (Rd x R m y ) uniformly compactly supported in R m y. Thus (u n ) can be seen as a bounded sequence of L 2 loc (Rd x; H), with H = L 2 (R m y ) and compactness modulo H means ( h L 2 (R m y )) u n (x, y)h(y)dy belongs to a compact set of L 2 loc (Rd x). This is exactly the goal of Averaging lemma.
Averaging lemma - Heterogeneous case The answer given by Gérard (Microlocal defect measures, Comm. Part. Diff. Eqs, 1991). Theorem 2. Let (u n ) be a bounded sequence in L 2 loc (Rd x R m y ; C r ) and x α A α (x, y)u n = y β g n, β N m 0, α k where (g n ) is relatively compact in H k loc (Rd x R m y ). If the principal symbol p(x, y, ξ) = A α (x, y)ξ α satisfies ( (x, ξ) R d x S d 1 ξ ) α =k p(x, y, ξ) is regular, (a.e. y R m ), then u n (x, y)ρ(y)dy is relatively compact in L 2 loc (Rd x; C r ) for any ρ C c (R m y ). Coefficients A α C(R d x; L (R m y ))
Velocity averaging a general framework P - a (fractional) differential operator α k x k, α k R + multiplier operator with the symbol (2πiξ k ) α k, A sequence of equations: Pu n (x, y) = d k=1 u n 0 in L 2 (R m y ; L p (R d x)), p 2 a k α k x k (a k (x, y)u n (x, y)) = β y g n (x, y) (1) { L (R m y ; C b (R d x)), p = 2 L (R m y ; L r (R d x)), 2/p + 1/r = 1, p > 2 g n 0 in L 2 (R m y ; W α,p (R d x)), α = (α 1,..., α d ), W α,s (R d ) is a dual of W α,s (R d ) = {u L s (R d ) : α k k u Ls (R d ), k = 1,..., d}.
The main result Theorem 3. Assume that u n 0 weakly in L 2 (R m ; L p (R d )) L 2 (R d+m ), p 2, where u n represent weak solutions to (1). For p = 2 we assume that for every (x, ξ) R d P A(x, ξ, y) := d a k (x, y)(2πiξ k ) αk 0 (a.e. y R m ). (2) k=1 If p > 2, the last assumption is reduced to almost every x R d and every ξ P. Then, for any ρ L 2 c(r m ), R m u n (x, y)ρ(y)dy 0 strongly in L 2 loc(r d ).
Generalised H-measures We consider smooth manifolds (instead of S d 1 ) P = {ξ R d : d ξ k lα k = 1}, k=1 l a minimal number such that lα k > d for each k. Denote by ( π P (ξ) = ξ 1 ( ) ξ lα1 1 + + ξ lα 1/lα1,..., d d a projection of R d \{0} on P. Corollary of the Marzinkiewicz multiplier theorem: Lemma 1. ξ d ( ξ lα1 1 + + ξ lα d d For any ψ C d (P), the composition ψ π P is an L p -multiplier, p 1,. ) 1/lαd ),
Generalised H-measures Theorem 4. (Existence) Let u n 0 in L 2 (R d ; C r ). Then, after extracting a subsequence, there exists a hermitian, positive semi-definite r r matrix Radon measure µ on R d P such that for all ϕ 1, ϕ 2 C 0 (R d ), ψ C(P) lim F(ϕ 1 u i n n )(ξ)f(ϕ 2u j n )(ξ) ψ π P(ξ)dξ = µ ij, ϕ 1 ϕ 2 ψ R d = ϕ 1 (x)ϕ 2 (x)ψ(ξ)dµ ij (x, ξ). R d P The measure µ we call the H P -measure corresponding to the (sub)sequence (of) (u n ).
The idea behind the proof Basic tool - generalised H-measures. Extended to functions taking values in infinite dimensional Banach spaces, i.e u n L 2 (R m ; L p (R d )), p 2 (nontrivial generalisation!). Bunch of auxiliary results dealing with their properties, representations etc. (skipped today!). Important - for p > 2 they are absolutely continuous with respect to the Lebesgue measure!
Sketch of the proof Let u n be weak solutions to (1), i.e.: R m+d k=1 d a k (x, p)u n (x, p)( xk ) α k (f(x, p))dxdp = ( 1) κ g n (, p), p κ f(, p) dp, (3) f W c κ,2 (R m ; W α,p (R d )) a test function. Take: f n (x, p) = ρ 1 (p) (I A ψ πp ) ( ϕu n (, q) ) (x)ρ 2 (q)dq, R m R m where ψ C d (P), ϕ C c (R d ), ρ 1, ρ 2 C κ c (R m ).
The idea behind the proof I the multiplier operator with the symbol where 1 θ(ξ) ( ξ 1 lα1 + + ξ d lα d ) 1/l, θ C c (R d ) a cut-off function, θ = 1 around the origin. I is a (generalised) version of the classical Riesz potential I k The following lemma is essential. Lemma 2. F(I k u)(ξ) = 1 ξ k û(ξ). The multiplier operator I : L 2 (R d ) L p (R d ) W α,p (R d ) is bounded (with L p norm considered on the domain), as well as I A ψ πp.
Sketch of the proof Due to the strong convergence of (g n ) by taking limit in (3): R 2m R d P A(x, ξ, p)ρ 1 (p)ρ 2 (q)ϕ(x)ψ(ξ)dµ(p, q, x, ξ)dpdq = 0, where A(x, ξ, p) = d k=1 (2πiξ k) α k a k. As the test functions ρ i, ϕ, and ψ are taken from dense subsets in appropriate spaces, we conclude A(x, ξ, p)dµ(p, q, x, ξ) = 0, (a.e. p, q R 2m ). The non-degeneracy condition (2) directly implies µ = 0. The very definition of µ gives ( ρ L 2 (R m )) u n (x, y)ρ(y)dy belongs to a compact set of L 2 loc (Rd ). Q.E.D.
An ultra-parabolic equation An ultra-parabolic equation with discontinuous coefficients in Ω R d : where div f(x, u) div div B(x, u) + ψ(x, u) = 0, (4) B(x, u) a symmetric matrix (b jk ) 0, min(j, k) l < d, while B = (b jk ) j,k=l+1,...,d satisfies an ellipticity condition ( ξ R d l ), y 1, y 2 R, x R d, (y 1 > y 2 ) = ( B(x, y 1 ) B(x, y 2 )) ξ ξ c ξ 2, c > 0; ψ C 1 (R y ; L (Ω)); y f k, y b jk L 2 loc (R y; L r loc (Ω)), r > 1.
Entropy solutions We say that a function u L (Ω) represents an entropy admissible weak solution to (4) if for every y R it holds div (sgn(u(x) y) ( f(x, u(x)) f(x, y) )) ( div div sgn(u(x) y) ( B(x, u(x)) B(x, y) )) = g(x, y) g(, y) a Radon measure on Ω.
Theorem 5. Assume that the coefficients of equation (4) satisfy the genuine non-degeneracy condition analogical to (2): for every ξ = (ˆξ, ξ) P = {(ˆξ, ξ) R l R d l : ˆξ 2 + ξ 4 = 1} and almost every x R d 2πi l ξ k y f k (x, y) + 4π 2 y B(x, y)ξ, ξ 0 (a.e. y R). k=1 Then, a sequence of entropy solutions (u n ) to (4) such that u n L (Ω) < M for every n N is strongly precompact in L2 loc (Ω).
The idea Consider the kinetic formulation of the ultra-parabolic equation (4) ( ) div h n (x, y) y f(x, y) div div (h n (x, y) y B(x, y)) = y g n (x, y) where h n (x, y) = sgn(u n (x) y). This is a special case of our general theory implying ( M M h n(x, y)dy) is strongly precompact in L 2 loc (Ω). Since the result follows. M M h n (x, y)dy = 2u n (x)
H-distributions If p < 2, H-measures are not appropriate (study the limit of u 2 n). Their extension to an L p L q setting is called H-distributions (Antonić, Mitrović, 2011). Theorem 6. (Existence) (u n ) bounded sequence of functions in L p (R d+m ), p 1, 2, (v n ) bounded sequence of uniformly compactly supported functions in L (R d+m ). Then, after passing to a subsequence (not relabelled), for any p 1, s there exists a continuous bilinear functional B on L p (R d+m ) C d (P) such that for every ϕ L p (R d+m ), ψ C d (P) B(ϕ, ψ) = lim ϕ(x, y)u n (x, y) ( ) A ψp v n (x, y)dxdy, n R d+m where A ψp ψ π P. is the (Fourier) multiplier operator on R d x associated to
On an extension of a bilinear functional on L p (R d ) E to a Bochner space In order to get a velocity averaging result, the H-distributions has to be applied to non-smooth functions with non-separated variables x and y. The Schwartz kernel theorem provides an extension as a distribution from D (R d+m P). We need a better result - extension on L p (R d+m ; C d (P)). Theorem 7. B continuous bilinear functional on L p (R d ) E, E separable Banach space, p 1,. Then B can be extended as a continuous functional on L p (R d ; E) if and only if there exists a (non-negative) function b L p (R d ) such that for every ψ E it holds Bψ(x) b(x) ψ E, (a.e. x R d ), where B is a bounded linear operator E L p (R d ) defined by Bψ, ϕ = B(ϕ, ψ), ϕ L p (R d ).
An application to the velocity averaging We consider a sequence of solutions u n to d α k x k (a k (x, y)u n (x, y)) = y β g n (x, y), k=1 where a) u n 0 in L p (R d+m ), p > 1. b) a k L p (R d+m ), for some p 1, p, k = 1,..., d, c) g n 0 in Lˆp (R m ; W α,ˆp (R d )), where α = (α 1,..., α d ) and 1/ p + 1/ˆp = 1/p. Theorem 8. Assume A 2 A 2 + δ 1 in L p loc (Rd+m ; C d (P)) strongly as δ 0. Then, for any ρ C c (R m ), the sequence R m ρ(y)u n (, y)dy strongly converges to zero in L p loc (Rd ).
Literature [AM] N. Antonić, D. Mitrović, H-distributions an extension of the H-measures in L p L( ) q setting, Abstr. Appl. Anal. 2011 (2011), 12 pp. [G] P. Gérard: Microlocal defect measures, Comm. Partial Differential Equations 16 (1991) 1761 1794 [LM1] M. Lazar, D. Mitrović, Velocity averaging a general framework, DynPDE 9 (2012), 239 260, [LM2] M. Lazar, D. Mitrović, On an extension of a bilinear functional on L p (R d ) E to a Bochner space with an application to velocity averaging, C. R. Acad. Sci. Paris, Ser. I, to appear [P] E. Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci. 159 (2009), 180 228. Thanks for your attention!