Degenerate parabolic stochastic partial differential equations

More documents

Recommendations

Info

M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 4301 Proof. In order to prove this claim, we denote by σ i the ith row of σ . Let us fix test functions ψ 1 ∈ C ∞ (T N ), ψ 2 ∈ Cc ∞(R) and define θ(ξ) = ξ −∞ ψ 2(ζ ) dζ . We denote by ⟨·, ·⟩ x the duality between the space of distributions over T N and C ∞ (T N ). It holds σ i v ∇g, ψ 1 ψ 2 = − div(σ i ψ 1 ), ψ 2 (ξ) dξ = − div σ i ψ 1 , θ(v) x If the following was true = σ i ∇θ(v), ψ 1 x . −∞ σ i ∇θ(v) = θ ′ (v) σ i ∇v in D ′ (T N ), (12) we would obtain σ i ∇g, ψ 1 ψ 2 = θ ′ (v) σ i ∇v, ψ 1 x = σ i ∇v δ v=ξ , ψ 1 ψ 2 and the proof would be complete. Hence it remains to verify (12). Towards this end, let us consider an approximation to the identity on T N , denoted by (ϱ τ ). To be more precise, let ˜ϱ ∈ Cc ∞(RN ) be a nonnegative symmetric function satisfying R N ˜ϱ = 1 and supp ˜ϱ ⊂ B(0, 1/2). This function can be easily extended to become Z N -periodic, and let this modification be denoted by ¯ϱ. Now it is correct to define ϱ = ¯ϱ ◦q −1 , where q denotes the quotient mapping q : R N → T N = R N /Z N , and finally ϱ τ (x) = 1 τ N ϱ x . τ Since the identity (12) is fulfilled by any sufficiently regular v, let us consider v τ , the mollifications of v given by (ϱ τ ). We have σ i ∇θ(v τ ) −→ σ i ∇θ(v) in D ′ (T N ). In order to obtain convergence of the corresponding right hand sides, i.e. θ ′ (v τ ) σ i ∇v τ −→ θ ′ (v) σ i ∇v in D ′ (T N ), we employ similar arguments as in the commutation lemma of DiPerna and Lions (see [10, Lemma II.1]). Namely, since σ i (∇v) ∈ L 2 (T N ) it is approximated in L 2 (T N ) by its mollifications [σ i ∇v] τ . Consequently, θ ′ (v τ ) σ i ∇v τ −→ θ ′ (v) σ i ∇v in D ′ (T N ). Thus, it is enough to show that θ ′ (v τ ) σ i ∇v τ − σ i ∇v τ −→ 0 in D ′ (T N ). (13) It holds σ i (x)∇v τ (x) − σ i ∇v τ (x) = v(y) σ i (x)(∇ϱ τ )(x − y) dy + v(y)div y σ i (y)ϱ τ (x − y) dy T N T N = − v(y) σ i (y) − σ i (x) (∇ϱ τ )(x − y)dy + v(y)div σ i (y) ϱ τ (x − y)dy. T N T N x
4302 M. Hofmanová / Stochastic Processes and their Applications 123 (2013) 4294–4336 The second term on the right hand side is the mollification of vdivσ i ∈ L 2 (T N ) hence converges in L 2 (T N ) to vdivσ i . We will show that the first term converges in L 1 (T N ) to −vdivσ i . Since τ|∇ϱ τ |(·) ≤ Cϱ 2τ (·) with a constant independent on τ, we obtain the following estimate v(y) σ i (y) − σ i (x) (∇ϱ τ )(x − y) dy ≤ C∥σ i ∥ W 1,∞ T N (T N ) ∥v∥ L 2 (T N ) . L 2 (T N ) Due to this estimate, it is sufficient to consider v and σ i smooth and the general case can be concluded by a density argument. We infer 4 − v(y) σ i (y) − σ i (x) (∇ϱ τ )(x − y) dy T N = − 1 τ N+1 v(y) Dσ i x + r(y − x) x − y (y − x) · (∇ϱ) dr dy τ T N 1 T N 1 0 = v(x − τ z) Dσ i (x − rτ z)z · (∇ϱ)(z) dr dz 0 −→ v(x) Dσ i (x) : z ⊗ (∇ϱ)(z) dz, T N ∀x ∈ T N . Integration by parts now yields z ⊗ (∇ϱ)(z) dz = −Id (14) T N hence v(x) Dσ i (x) : z ⊗ (∇ϱ)(z) dz = −v(x)div σ i (x) , ∀x ∈ T N , T N and the convergence in L 1 (T N ) follows by the Vitali convergence theorem from the above estimate. Employing the Vitali convergence theorem again, we obtain (13) and consequently also (12) which completes the proof. □ We proceed by two related definitions, which will be useful especially in the proof of uniqueness. Definition 2.6 (Young Measure). Let (X, λ) be a finite measure space. A mapping ν from X to the set of probability measures on R is said to be a Young measure if, for all ψ ∈ C b (R), the map z → ν z (ψ) from X into R is measurable. We say that a Young measure ν vanishes at infinity if, for all p ≥ 1, |ξ| p dν z (ξ) dλ(z) < ∞. X R Definition 2.7 (Kinetic Function). Let (X, λ) be a finite measure space. A measurable function f : X × R → [0, 1] is said to be a kinetic function if there exists a Young measure ν on X vanishing at infinity such that, for λ-a.e. z ∈ X, for all ξ ∈ R, f (z, ξ) = ν z (ξ, ∞). 4 By : we denote the component-wise inner product of matrices and by ⊗ the tensor product.
Page 1 and 2: Available online at www.sciencedire
Page 3 and 4: 4296 M. Hofmanová / Stochastic Pro
Page 7: 4300 M. Hofmanová / Stochastic Pro
Page 43: 4336 M. Hofmanová / Stochastic Pro

Degenerate parabolic stochastic partial differential equations

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?