Linear and Nonlinear Waves in Gas Dynamics Barbara Lee Keyfitz The - - PowerPoint PPT Presentation

Linear and Nonlinear Waves in Gas Dynamics

Barbara Lee Keyfitz

The Ohio State University bkeyfitz@math.ohio-state.edu

April 2, 2016

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 1 / 28

Outline

1 Background

Characteristics

2 The Problem

Explanation and People

3 Wave Equations

Basic Linear Quasilinear: Gas Dynamics

4 Compressible Gas Dynamics System

Self-Similar Solutions Free Boundary Problem and Local Solution Mysteries and Constraints

5 Recent Results

Incompressible Flow: Himonas & Misio lek Compressible Flow

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 2 / 28

Background Characteristics

An Example

PDE ∂u ∂t + a∂u ∂x = 0 General solution (f ∈ C1 classical solution; f ∈ D′ weak solution) u(x, t) = f (x − at) Characteristic curves x − at = const are significant: propagation of signals separation of regions of smooth flow unsuitable for prescribing data We see geometry in R2 = (x, t): tangents to char are t = (a, 1) geometry in dual space R2: char normals ν = (1, −a) = (ξ, τ) satisfy τ + aξ = 0; recall principal symbol of ∂t + a∂x is τ + aξ

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Background Characteristics

What You Saw Was an Example of

A first-order system:

d

• Aj

∂u ∂xj + b = 0 Aj are n × n matrices, u, b are n-vectors L0 = d

0 Ajξj is the principal symbol (matrix),

soln ν = (ξ0, . . . , ξd) to det ( Ajξj) = 0 is characteristic normal surfaces in Rd+1 = {(x0, . . . , xd)} whose normals are characteristic are characteristic surfaces we will have different stories for Aj constant, Aj = Aj(x) (linear system) and Aj = Aj(x, u) (quasilinear system)

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Background Characteristics

Significance

Cauchy-Kovalevsky Theorem: locally, and for analytic functions, data

• n a non-characteristic surface ⇒ ∃! analytic solution

Classification of PDE via characteristics:

elliptic (no real characteristics) hyperbolic (maximal set of real characteristics) ‘parabolic’: borderline, in between, mixed, dispersive etc

Hyperbolic system (with distinguished time variable) A0 ∂u ∂t +

d

• 1

Aj ∂u ∂xj + b = 0

Aj = Aj(x, t, u), b = b(x, t, u) (quasilinear) det (τI + A−1

0 Ajξj) = 0 ⇒ τi(ξ) real (characteristic normals)

Symmetrizable hyperbolic: A0 symm, pos def; Aj symm, ∀j

Cauchy problem for linear hyperbolic systems well-posed in Hs

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 5 / 28

The Problem Explanation and People

The Focus of this Talk: Quasilinear Systems

We are faced with some incompatible difficulties Local (=short time) existence for quasilinear (QL) symmetrizable hyperbolic systems in Hs, s > d/2 + 1 Life-span of Hs solutions depends on u0s (and on Aj) Solutions of QL systems don’t stay in Hs (e.g., Burgers Equation): ut + uux = 0 , u(x, 0) = u0 , u(x + u0(x)t, t) = u0(x) , t < − 1 u′

0(x)

If d = 1 and u0BV < δ0, ∃! global (in t) solution in BV (= W 1,∞) for QL systems ∂tu +∂xf (u) = 0 with A(u) = df strictly hyperbolic + For d ≥ 2, only theory is short-time result in Hs; no other theory Theorem (Littman, Brenner, Rauch): Do not expect well-posedness in W m,p for p = 2 Our idea: explore what goes wrong for some examples

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 6 / 28

The Problem Explanation and People

People Working in this Direction

My co-authors Suncica Canic John Holmes Gary Lieberman David Wagner Eun Heui Kim Allen Tesdall Katarina Jegdic Hao Ying Feride Tiglay Other people and groups (incomplete list) Gui Qiang Chen Mikhail Feldman Tai Ping Liu Volker Elling Yuxi Zheng Marshall Slemrod Dehua Wang

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The Problem Explanation and People

Target Equations

Equations of Ideal Compressible Gas Dynamics in Two Space Dimensions System of 4 equations Important and well-studied (engineering and computation) Comparison possible with simplified models Initial approach looked at selfsimilar solutions (x t and y t ) Leads to tractable problems (and insight into what might go wrong) Self-similar approach ⇒ interesting analysis Also look beyond self-similar problems

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 8 / 28

Wave Equations Basic Linear

Warm-up Problem: Linear Wave Equation

Example: Wave eqn in 2-D: utt = c2(uxx + uyy) As a system u1 = uy , u2 = ut − cux ∂t u1 u2

• = c

1 −1

• ∂x

u1 u2

• + c

1 1

• ∂y

u1 u2

• Normals

ν = (ξ, η, ±c

• ξ2 + η2);

Char vbles v =

• η

ξ ∓

• ξ2 + η2
• Characteristic variables: eigenvectors {v = v(

ν) ∈ Rn | L0v = 0} Characteristic normals: τ 2 = c2(ξ2 + η2), the “characteristic cone” Characteristic surfaces: planes ξx + ηy ± c

• ξ2 + η2t = 0

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 9 / 28

Wave Equations Basic Linear

Fundamental Solution to Linear Wave Equation

Envelope of characteristic surfaces through (0, 0, 0) is the “wave cone”: x2 + y2 = c2t2 Boundary of domain of influence of the origin Support cone of fundamental solution (support includes interior) Singular support is boundary of cone Data u(x, y, 0) = 0, ut(x, y, 0) = u0(x, y): u(t, x, y) = 1 4πc

• B

u0(ξ, η)

• c2t2 − (x − ξ)2 − (y − η)2 dξ dη = KW ∗ u0

where B = {(ξ, η) | (x − ξ)2 + (y − η)2 ≤ c2t2} KW (·, t) ∈ H−d/2+1−ε

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 10 / 28

Wave Equations Quasilinear: Gas Dynamics

Characteristic Structure of Gas Dynamics Equations

Full Euler Equations: Adiabatic, compressible, ideal gas dynamics “conserved quantities” “fluxes” ρt + (ρu)x + (ρv)y = 0 (ρu)t + (ρu2 + p)x + (ρuv)y = 0 (ρv)t + (ρuv)x + (ρv2 + p)y = 0 (ρE)t + (ρuH)x + (ρvH)y = 0 State variables ρ (density), (u, v) (velocity), and p (pressure) E = 1 γ − 1 p ρ + 1 2(u2 + v2), H = γ γ − 1 p ρ + 1 2(u2 + v2) γ ≈ 1.4: parameter (ratio of specific heats) Linearize at a state u = (ρ, u, v, p) Roots of characteristic equation (det L0 = 0) – real; a double root: ¯ τ = 0, 0, ±

• γp/ρ
• ξ2 + η2 , where ¯

τ = τ − (ξu + ηv)

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Wave Equations Quasilinear: Gas Dynamics

Characteristic Normals, Familiar and Unfamiliar

Four families fall into two types; fix u = (ρ, u, v, p) to study. (1) τ = (ξu + ηv): all normals lie in a plane with normal (u, v, −1). Corresponding char surfaces all contain (u, v, −1) (their envelope) Dynamic behavior is “transport” equation Domain of influence of (0, 0, 0) is the line (−ut, −vt, t). (2) pair τ = (ξu + ηv) ±

• γp/ρ
• ξ2 + η2

Normals form pair of conical surfaces (forward and backward) like WE Wave cone, envelope of corresponding char surfaces, is tilted cone (tilt depends on u, v) with size (“speed”) depending on p and ρ) Note (1) is also a double characteristic Contrast (1) and (2) in two ways: “transport” vs “wave equation” and “nonlinear” (propagation speed depends on states) vs “linearly degenerate” (∇τ(u, ξ) · r(u, ξ) = 0)

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 12 / 28

Wave Equations Quasilinear: Gas Dynamics

Summary of the Structure

y t x

Characteristic Normals Pair: τ = (ξu+ηv)±

• γp/ρ
• ξ2 + η2

– acoustic waves – “wave-equation-like” propagation – genuinely nonlinear (Burgers)

Pair: τ = ξu + ηv (double)

– entropy and vorticity waves – transport-equation-like propagation – linearly degenerate

For general QL systems, other possibilities exist, but they may not correspond to anything that occurs in physical models

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Compressible Gas Dynamics System Self-Similar Solutions

Self-similar Problems: 2-D Riemann Problems (with ˇ Cani´ c, Lieberman, Kim, Jegdic, Tesdall, Popivanov, Payne, Ying)

• Analogy with 1-D: focus on transport and wave interactions
• Benchmark problem: Shock reflection by a wedge

X= t Ξ S= t Σ

Flow Wedge

Incident Shock Reflected Shock t<0 t=0 t>0

• Work in self-similar coordinates: ξ = x

t , η = y t

• Reduced eq’n (−ξ + A(U))Uξ + (−η + B(U))Uη = 0
• Type Changes: hyperbolic for (ξ, η) >> 1; ‘subsonic’ region near 0
• Change of Type in Nondegenerate Waves Only

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 14 / 28

Compressible Gas Dynamics System Self-Similar Solutions

Reformulation of Self-similar Equations

Subsonic ⇒ 2 complex characteristics ⇒ “Elliptic” part ⇒ express with a second-order equation that changes type at the sonic line: (u − ξ)2 + (v − η)2 ≡ U2 + V 2 = c2 ≡ c2(ρ, p)

• r

c2(ρ) Simplified isentropic equations (full system is similar) Q(ρ; U, V ) = (c2 − U2)ρξξ − 2UV ρξη + (c2 − V 2)ρηη + 2cc′(ρ2

ξ + ρ2 η)

− 2ρξ

• U(1 + Uξ + Vη) − c2 U(Vη+1)−VVξ

U2+V 2

• − 2ρη
• V (1 + Uξ + Vη) + c2 UUη−V (Uξ+1)

U2+V 2

• Transport equations for u and v

(U, V ) · ∇U + U = −pξ/ρ = −c2ρξ/ρ ≡ qξ(ρ) (U, V ) · ∇V + V = −pη/ρ = −c2ρη/ρ ≡ qη(ρ)

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Compressible Gas Dynamics System Free Boundary Problem and Local Solution

Local Solution for Regular Reflection

Elliptic FBP: Q(ρ; U, V ) = 0 elliptic in Ω (“acoustic” characteristics)

η

Σ Σ0 Ω σ

Ξs=(ξs,0) Ξ0=(ξ0,0) Ξ1

Boundary conditions: ρη = 0 (symmetry) on Σ0 N(ρ) ≡ β · ∇ρ = 0 (oblique deriv) on Σ Compatibility ρ = ρs at Ξs; Cutoff ρ = f on σ (cutoff boundary) Transport (linear, degenerate characteristic) (U, V ) · ∇U + U = −pξ/ρ , (U, V ) · ∇V + V = −pη/ρ U|Σ = U0, V |Σ = V0 from Rankine-Hugoniot equations Free boundary : Shock evolution: dη

dξ = F(ρ, U, V ) also from RH equations

Theorem BC at Ξs cannot be prescribed BUT ∃ choices of f that give ρ(Ξs) = ρs.

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Compressible Gas Dynamics System Free Boundary Problem and Local Solution

The Elliptic Problem

Approach: FPT (CK & Lieberman, CKK) in weighted H¨

• lder spaces

Linear: L ρ = aij(w, W , Z)∂ijρ + bi(w, W , Z, ∇W , ∇Z)∂iρ = 0 in Ω M ρ = β(w, W , Z) · ∇ρ = 0 on Σ, ρη = 0 on Σ0, ρ(η) = f (η) on σ Theorem For each (w, W , Z), ∃ choices of f that give a solution with ρ(Ξs) = ρs.

ξ η

Σ Σ0 Ω σ

Ξs=(ξs,0) Ξ0=(ξ0,0) Ξ1

Solve FBP by fixing boundary and iterating using shock evolution equation. Proof Theory of Oblique Der and mixed problems (Lieberman) Theorem For each (W , Z), the mapping w → ρ has a fixed point. Proof Compactness of solution operator of linear elliptic eqn

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Compressible Gas Dynamics System Free Boundary Problem and Local Solution

The Transport Problem

Theorem The mapping (W , Z) → (U, V ) is a contraction Reformulate (parameter defined by the dynamics) as ˙ ξ = U , ˙ η = V , ˙ U = −U − q(ρ)ξ or ˙ U = −U − Q1(ξ, η) ˙ V = −V − q(ρ)η or ˙ V = −V − Q2(ξ, η) Nbhd of Ξs: U ≤ a0 < 0 so simplify to

η

(U0(θ),V0(θ)) Σ=Γ(θ) Σ0 σ

Ξs=(ξs,0) Ξ0=(ξ0,0) Ξ1

dη dξ = V U dU dξ = −1 − Q1(ξ, η) U dV dξ = −V U − Q2(ξ, η) U

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 18 / 28

Compressible Gas Dynamics System Free Boundary Problem and Local Solution

Important Point: “ρ hardly depends on (W , Z)”

Theorem For ρ1 = ρ[W1, Z1] and ρ2 = ρ[W2, Z2], we have

• ρ1 − ρ2
• (−(γ+1))

2+ǫ

≤ M

• W1 − W2
• (−γ)

1+ǫ +

• Z1 − Z2
• (−γ)

1+ǫ

• H¨
• lder norm weighted at corners and elliptic estimates for ρ1 − ρ2

bounds near Σ since (W1, Z1) = (W2, Z2) at Σ contraction ⇒ fixed point ⇒ solution to fixed boundary problem Main Theorem (Jegdi´ c, Keyfitz, ˇ Cani´ c and Ying) In a (small) region behind the reflection point, there is a solution to the self-similar equations, and a corresponding position for the reflected shock. Proof Shock evolution eqn ⇒ updated shock η; Tη = η compact; fixed point

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 19 / 28

Compressible Gas Dynamics System Mysteries and Constraints

Nonlinear Behavior We Do Not Yet Understand

x/t

1.0746 1.0748 1.075 1.0752 1.0754 1.0756 0.41 .4102 .4104 .4106 .4108

Discovered in numerical simulations (in UTSD, NLWS and Full Euler system); verified experimentally by B. W. Skews & al. (JFM) No theory as yet (but definitely involves acoustic waves)

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 20 / 28

Compressible Gas Dynamics System Mysteries and Constraints

Moving On: What Can Be Expected in Lp

Walter Littman (1963): Wave equation

d

• 1

∂2u ∂x2

i

− ∂2u ∂t2 = 0 Energy in Lp: Ep(t) = d

• 1
• ∂u

∂xi

• p

+

• ∂u

∂t

• p

dx Theorem An estimate Ep(t) ≤ C(t)Ep(0) holds only if either p = 2 or d = 1. Philip Brenner (1966): ∂u

∂t = d 1 Aj ∂u ∂xj + Bu

Theorem Suppose p = 2, 1 ≤ p ≤ ∞. The Cauchy problem is well-posed in Lp if and only if the matrices A1, . . . , Ad commute. Note: Commuting matrices ⇒ all char families are of transport type. Rauch (1986): “BV Estimates Fail for Most Quasilinear Hyperbolic Systems in Dimensions Greater Than One”

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Recent Results Incompressible Flow: Himonas & Misio lek

Classical Solutions, Result of Himonas-Misio lek

‘Incompressible Euler’ (incompressible fluid in n-D domain Ω) ∂tu + ∇uu + ∇p = 0 ∇ · u = 0 u(x, 0) = u0(x) , x ∈ Ω , t > 0 . Classical well-posedness. Theorem (Classical) If s > d/2 + 1 and u0 ∈ Hs(Ω, Rd) with ∇ · u0 = 0, then ∃T > 0 and ∃!u ∈ C([0, T], Hs(Ω, Rd)) which depends ctsly on u0. We have u(t)Hs ≤ u0Hs 1 − Ctu0Hs cf: “nonphysical” weak solutions of DeLellis and Sz´ ekelyhidi

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Recent Results Incompressible Flow: Himonas & Misio lek

Result for Periodic Data

Theorem (Himonas-Misio lek) Let Ω = T2. The solution map, u0 → u, is not uniformly cts from the unit ball in Hs(Ω, R2) into C([0, T], Hs(Ω, R2)). Specifically, take data u1,n and u−1,n with u1,n − u−1,n s ≃ 1

n:

u1,n

0 (x, y) =

1 n + 1 ns cos ny, 1 n + 1 ns cos nx

• u−1,n

(x, y) =

• −1

n + 1 ns cos ny, −1 n + 1 ns cos nx

• and exact solution (periodic case T2):

u1,n(x, y, t) = 1 n + 1 ns cos(ny − t), 1 n + 1 ns cos(nx − t)

• u−1,n(x, y, t) =
• −1

n + 1 ns cos(ny + t), −1 n + 1 ns cos(nx + t)

• Barbara Keyfitz (Ohio State)

Linear and Nonlinear Waves April 2, 2016 23 / 28

Recent Results Incompressible Flow: Himonas & Misio lek

Calculation

Proof. u1,n − u−1,n Hs ≃ 1 n but u1,n − u−1,nHs ≃ | sin t| − 1 n Theorem (Himonas-Misio lek) With cut-off functions, get approximate solutions and similar result on R2 Also a similar result holds for R3

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 24 / 28

Recent Results Compressible Flow

Compressible Flow: Feride Tı˘ glay and John Holmes

We find an identical result in T2 (two space dim, periodic data) for the compressible Euler system ρt + (ρu)x + (ρv)y = 0 (ρu)t + (ρu2 + p)x + (ρuv)y = 0 (ρv)t + (ρuv)x + (ρv2 + p)y = 0 (ρE)t + (ρuE + up)x + (ρvE + vp)y = 0 Replace p by T = p/ρ and ignore conservation form (for classical solutions): ρt + (uρ)x + (vρ)y = 0 ut + uux + vuy + Tx + Tρx/ρ = 0 vt + uvx + vvy + Ty + Tρy/ρ = 0 Tt + uTx + vTy + (γ − 1)T(ux + vy) = 0

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 25 / 28

Recent Results Compressible Flow

Comparison of Compressible and Incompressible Equations

For adiabatic (polytropic) gases – E = 1

2(u2 + v2) + p (γ−1)ρ:

ρt + uρx + vρy + ρ(ux + vy) = 0 ut + uux + uvy + px/ρ = 0 vt + uvx + vvy + py/ρ = 0 pt + upx + vpy + γp(ux + vy) = 0 Compare with incompressible system ut + u · ∇u + ∇p = 0 , ∇u = 0 the “low Mach number limit” of compressible system (c → ∞): ux + vy = 0 ut + uux + uvy + px = 0 vt + uvx + vvy + py = 0

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 26 / 28

Recent Results Compressible Flow

Our H-M Imitation

Approximate solutions, for ω = 1 and ω = −1, periodic case: ρω,n ≡ ρ0 uω,n = ω n + 1 ns cos(ny − ωt) vω,n = ω n + 1 ns cos(nx − ωt) T ω,n = T0 + 1 n2sρ0 sin(nx − ωt) sin(ny − ωt) Use short-time well-posedness result (Kato, Lax, Beale-Majda): Theorem For s > 2 (i.e. d/2 + 1), and T > 0 depending on u0s, ∃! solution into C[(0, T), Hs(Ω, R2)].

Barbara Keyfitz (Ohio State) Linear and Nonlinear Waves April 2, 2016 27 / 28

Recent Results Compressible Flow

Our results

Theorem (K-Tı˘ glay, Holmes-K-Tı˘ glay) For the given data (periodic or in the plane), the exact solution(s) uω,n are Hs-close to the approximate solutions uω,n and so u1,n − u−1,nHs ≃ | sin t| − 1 n Interesting sidenote: Kato claims uniformly continuous dependence. Theorem (Kato) If at t = 0, un

0 − v0s ≤ M and un 0 − v00 → 0 as n → ∞, then

un(t) − v(t)s−1 → 0 uniformly in t No foul: A different notion of uniformity. We (and H-M) do not have a v! Even though we are tracking linear waves, we are using the nonlinearity of the system

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