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Burgers Equation (Burgers ’48)
Viscous Burgers Equation (BE):
∂u ∂t + (u · ∇) u
- convection
= ν∆u, ν > 0 (BE) describes laminar flow in fluid dynamics.
Aim:
- 1. Formulation on SG
- 2. Existence of solutions
The viscous Burgers Equation on the Sierpinski gasket Melissa - - PowerPoint PPT Presentation
The viscous Burgers Equation on the Sierpinski gasket Melissa - - PowerPoint PPT Presentation
The viscous Burgers Equation on the Sierpinski gasket Melissa Meinert Bielefeld University 6th Cornell Conference on Analysis, Probability and Mathematical Physics on Fractals Cornell University June 16, 2017 Burgers Equation (Burgers 48)
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Starting point
Consider the Cauchy problem for the Heat Equation (HE):
- wt(x, t) = ν∆w(x, t),
t > 0 w(x, 0) = w0(x) with infx∈SG w0(x) > c0 > 0, w0 ∈ C(SG). Idea: Use knowledge about (HE) and Cole Hopf Transformation [Col51, Hop50] u(x, t) := −2ν (w(x, t))x w(x, t) to proof existence of solutions!
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Setup
◮ X = SG Sierpinski gasket ◮ µ finite Borel measure s.t. µ(U) > 0 ∀U ⊂ SG open, U = ∅ ◮ (E, F) standard resistence form ◮ ∆µ Laplacian, defined by
E(u, v) = −
- v∆µudµ
for all v ∈ F vanishing on the boundary See Kusuoka [Kus89], Kigami [Kig89, Kig93, Kig01] and Strichartz [Str06].
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Remark
resistance form + SG compact in resistance ⇒ F ⊂ C(SG) only contains bounded functions F algebra, see [BH91, Cor. I.3.3.2], and it holds E(fg)
1 2 ≤ f ∞E(g) 1 2 + g∞E(f ) 1 2
∀ f , g ∈ F Following [HRT13], see also [IRT12], we use the framework of 1-forms and derivations introduced by Cipriani and Sauvageot [CS03]. F ⊗ F :=
- i
fi ⊗ gi finite linear combination, fi, gi ∈ F
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Definition ([CS03])
On F ⊗ F let a ⊗ b, c ⊗ dH = 1 2 (E(a, cdb) + E(abd, c) − E(bd, ac)) and · H the associated semi-norm. The space H is obtained by taking the quotient by the normzero subspace and then the completion in · H. a (b ⊗ c) = (ab) ⊗ c − a ⊗ (bc) (b ⊗ c) d = b ⊗ cd a, b, c, d ∈ F
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Theorem
[IRT12, Thm. 5.6] If H is the Hilbert module of a resistance form
- n a finitely ramified cell structure, then ∞
n=0 Hn is dense in H,
with the subspaces Hn =
- w∈Wn
hw ⊗ ✶Kw , where the sum is over all n-cells, ✶Kw is the indicator of the n-cell Kw, and hw is a n-harmonic function modulo additive constants. Further, if P is the projection from H onto the closure of the image of the derivation ∂, then
∞
- n=0
- w∈Wn
hw ⊗ ✶Kw = f ⊗ ✶, where f is n-harmonic is dense in PH.
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Definition (abstract derivation)
A derivation operator ∂ : F → H can be defined by setting ∂f := f ⊗ ✶, f ∈ F. It obeys the Leibniz property ∂(ab) = (∂a)b + a(∂b). and is a bounded linear operator satisfying ∂f 2
H = E(f ),
f ∈ F.
Remark
The operator ∂ : F → H can be extended to a closed linear
- perator ∂µ : L2(SG, µ) → H with domain dense in F. For f ∈ F
and F ∈ C 1(❘) the chain rule is also satisfied ∂F(f ) = F ′(f )∂f .
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In our case: E(f ) = lim
m→∞
5 3 m
p∈Vm
- q∼mp
(f (p) − f (q))2 with domain F g∂f 2
H = f ⊗ g2 H
= lim
m→∞
5 3 m
p∈Vm
- q∼mp
g(p) + g(q) 2 2 (f (p) − f (q))2
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Definition (divergence)
The divergence ∂∗
µ : H → L2(SG, µ) is defined as -adjoint operator
to ∂µ, equipped with the domain D(∂∗
µ) :=
- v ∈ H : ∃u ∈ L2(SG, µ) : u, φL2(SG,µ) = −v, φH ∀φ ∈ F
- .
For v ∈ D(∂∗
µ) set ∂∗ µv := u.
Remark
For f ∈ D(∆µ) the following is true: ∂µf ∈ D(∂∗
µ)
and ∆µf = ∂∗
µ∂µf .
We will consider f ∈ D(∆µ) such that ∆µf ∈ C(SG) ⊂ L2(SG, µ) (1)
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Definition
We denote with DH→C(SG)(∂∗
µ) :=
- v ∈ D(∂∗
µ) : ∂∗ µv ∈ C(SG)
- the space of test vector fields.
Further, for u ∈ H, v ∈ DH→C(SG)(∂∗
µ) we define
- ∂µ∂∗
µu
- (v) := −(∂∗
µu)(∂∗ µv),
∂µu, uH := −(∂∗
µv)u, uH.
For f as in (1), we have ∂f ∈ DH→C(SG)(∂∗
µ).
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Existence of weak solution
Definition
Let u0 ∈ H. We say that a function u : [0, ∞) → H with initial condition u0 is a weak solution of the abstract Burgers Equation, if the function is differentiable on (0, ∞) and obeys for all v ∈ DH→Cb(∂∗
µ)
- ∆µu(v) − ∂µu(t), u(t)H(v)
= ut(t), vH, t > 0 limt→0u(t) − u0, vH = 0. (2)
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Existence of weak solution
Definition
Let u0 ∈ H. We say that a function u : [0, ∞) → H with initial condition u0 is a weak solution of the abstract Burgers Equation, if the function is differentiable on (0, ∞) and obeys for all v ∈ DH→Cb(∂∗
µ)
- ∆µu(v) − ∂µu(t), u(t)H(v)
= ut(t), vH, t > 0 limt→0u(t) − u0, vH = 0. (2)
Theorem
Let w0 ∈ Cb a positively function with w0(p) ≥ c0, p ∈ SG, for a fixed constant c0 > 0.Then the function u : [0, ∞) → H, u(t) := −∂µ(log w(t)), t > 0, with u0 = −∂µ log w0 is a weak solution of the initial problem (2).
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References I
[BH91] Nicolas Bouleau and Francis Hirsch. Dirichlet forms and analysis on Wiener space, volume 14 of De Gruyter studies in mathematics ; 14. de Gruyter, Berlin [u.a.], 1991. [Col51] Julian D Cole. On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of applied mathematics, 9(3):225–236, 1951. [CS03] Fabio Cipriani and Jean-Luc Sauvageot. Derivations as square roots of dirichlet forms. JOURNAL OF FUNCTIONAL ANALYSIS, 2003, 201, 1, 78, 2003. [Hop50] Eberhard Hopf. The partial differential equation u sub t + uu sub x = mu sub xx. DTIC AND NTIS, 1950. [HRT13] Michael Hinz, Michael Rockner, and Alexander Teplyaev. Vector analysis for dirichlet forms and quasilinear pde and spde on metric measure spaces. STOCHASTIC PROCESSES AND THEIR APPLICATIONS, 2013, 123, 12, 4373, 2013. [IRT12] Marius Ionescu, Luke G. Rogers, and Alexander Teplyaev. Derivations and dirichlet forms on fractals. JOURNAL OF FUNCTIONAL ANALYSIS, 2012, 263, 8, 2141, 2012.
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