On the motion of compressible inviscid fluids driven by stochastic - PowerPoint PPT Presentation

On the motion of compressible inviscid fluids driven by stochastic forcing Eduard Feireisl based on joint work with D.Breit (Edinburgh), M.Hofmanov´ a (Berlin) Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague Irregular transport: Analysis and applications, Basel, 26 June – 30 June, 2017 The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC Grant Agreement 320078

Driven Euler system Field equations d ̺ + div x ( ̺ u ) d t = 0 d ( ̺ u ) + div x ( ̺ u ⊗ u ) d t + ∇ x p ( ̺ ) d t = ̺ G ( ̺, ̺ u ) d W , Stochastic forcing ∞ � ̺ G ( ̺, ̺ u ) d W = ̺ G k ( ̺, ̺ u ) d W k k =1 Iconic examples ∞ � ̺ G ( ̺, ̺ u ) d W = ̺ G k ( x ) d W k , ̺ G ( ̺, ̺ u ) d W = λ̺ u d W k =1

Data, initial and boundary conditions (Random) initial data ̺ (0 , · ) = ̺ 0 , ( ̺ u )(0 , · ) = ( ̺ u ) 0 W ≈ { W k } ∞ k =1 mutually independent Wiener processes Periodic boundary conditions � � N , N = (1) , 2 , 3 Ω = T N = [0 , 1] | { 0 , 1 }

Concepts of solutions Strong solution Solutions are smooth in space, spatial derivatives exist in the classical sense. Equations satisfied for Itˆ o’s stochastic integral Weak (PDE) solution Spatial derivatives understood in the sense of distributions Weak martingale solution Spatial derivatives understood in the sense of distributions. Data understood in terms of stochastic distribution - law. u 0 , W ∼ � ̺ 0 ∼ � ̺ 0 , u 0 ∼ � W Dissipative martingale solution Martingale solutions satisfying a suitable form of energy inequality

Weak (PDE) formulation Field equations �� � t = τ � τ � = ̺ u · ∇ x φ d x d t , ̺φ d x Ω t =0 0 Ω �� � t = τ � τ � ̺ u · φ d x − ̺ u ⊗ u : ∇ x φ + p ( ̺ ) div x φ d x d t Ω 0 Ω t =0 � τ �� � = ̺ G · φ d x d W 0 Ω φ = φ ( x ) − a smooth test function Stochastic integral (Itˆ o’s formulation) � τ �� � � τ �� � ∞ � ̺ G · φ d x d W = ̺ G k · φ d x d W k 0 Ω 0 Ω k =1

Admissibility - dissipative solutions Energy inequality � T � � � � 1 � 2 ̺ | u | 2 + H ( ̺ ) − ∂ t ψ d x d t 0 Ω � � | ( ̺ u ) 0 | 2 � ≤ ψ (0) + H ( ̺ 0 ) d x 2 ̺ 0 Ω � � � � T � T � | G k ( ̺, ̺ u ) | 2 +1 d t + ψ d x ψ d M E 2 ̺ 0 Ω 0 k ≥ 1 � ̺ p ( z ) ψ ≥ 0 , ψ ( T ) = 0 , H ( ̺ ) = ̺ d z z 2 1

Relative energy inequality Relative energy � � � � � 1 � � 2 ̺ | u − U | 2 + H ( ̺ ) − H ′ ( r )( ̺ − r ) − H ( r ) E ̺, u � r , U = d x Ω Relative energy inequality � T � � � � − ∂ t ψ E ̺, u � r , U d t 0 � T � T � � � � � � � � ≤ ψ (0) E ̺, u � r , U (0) + ψ d M RE + ψ R ̺, u � r , U d t 0 0 Test functions d r = D d t r d t + D s t r d W , d U = D d t U d t + D s t U d W

Remainder Remainder term � � � � � � � D d R ̺, u � r , U = t U + u · ∇ x U ( U − u ) d x ̺ Ω � � � ( r − ̺ ) H ′′ ( r ) D d t r + ∇ x H ′ ( r )( r U − ̺ u ) + d x Ω � − div x U ( p ( ̺ ) − p ( r )) d x Ω � � � � +1 � G k ( ̺, ̺ u ) 2 � � − [ D s ̺ t U ] k � d x 2 ̺ Ω k ≥ 1 � � � � +1 t r ] k | 2 d x + 1 t r ] k | 2 d x ̺ H ′′′ ( r ) | [ D s p ′′ ( r ) | [ D s 2 2 Ω Ω k ≥ 1 k ≥ 1

Existence theory Local existence of strong solutions [Kim [2011]], [Breit, EF, Hofmanov´ a [2017]] If the initial data are smooth, then the problem admits local-in-time smooth solutions. Solutions exist up to a (maximal) positive stopping time . The life-span is a random variable. Weak–strong uniqueness [Breit, EF, Hofmanov´ a [2016]] Pathwise uniqueness. A weak and strong solutions defined on the same probability space and emanating from the same initial data coincide as long as the latter exists Uniqueness in law. If a weak and strong solution are defined on a different probability space, then their laws are the same provided the laws of the initial data are the same

Weak (PDE) solutions Infinitely many weak (PDE) solutions, Breit, EF, Hofmanov´ a [2017] Let T > 0 and the initial data ̺ 0 ∈ C 3 (Ω) , ̺ 0 > 0 , u 0 ∈ C 3 (Ω) be given. There exists a sequence of strictly positive stopping times τ M > 0 , τ M → ∞ a.s. such that the initial–value problem for the compressible Euler system possesses infinitely many weak (PDE) solutions defined in (0 , T ∧ τ M ). Solutions are adapted to the filtration associated to the Wiener process W .

Semi-deterministic approach - additive noise “Additive noise” problem ∂ t ̺ + div x ( ̺ u ) = 0 ∞ � ∂ t ( ̺ u ) + div x ( ̺ u ⊗ u ) + ∇ x p ( ̺ ) = ̺ G k d W k k =1 � ∞ ̺ G k d W k = ̺ G d W k =1

Additive noise, Step I Step I ∂ t ( ̺ u − ̺ G W )+ div x ( ̺ u ⊗ u )+ ∇ x p ( ̺ ) = − ∂ t ̺ G W = div x ( ̺ u ) G W Transformed system I w = ̺ u − ̺ G W ∂ t ̺ + div x ( w + ̺ G W ) = 0 � ( w + ̺ G W ) ⊗ ( w + ̺ G W ) � ∂ t w + div x + ∇ x p ( ̺ ) ̺ = div x ( w + ̺ G W ) G W

Additive noise, Step II Step II � w = v + V + ∇ x Φ , div x v = 0 , v d x = 0 , V = V ( t ) Ω Transformed system II w = ̺ u − ̺ G W ∂ t ̺ + div x ( ∇ x Φ + ̺ G W ) = 0 � ( v + V + ∇ x Φ + ̺ G W ) ⊗ ( v + V + ∇ x Φ + ̺ G W ) � ∂ t v + div x ̺ + ∇ x p ( ̺ ) + ∇ x ∂ t Φ = div x ( ∇ x Φ + ̺ G W ) G W − ∂ t V

Additive noise, Step III Step III Fix Φ, ̺ , V so that � ̺ (0 , · ) = ̺ 0 , V (0) = 1 u 0 d x , ∇ x Φ(0 , · ) = H ⊥ [ u 0 ] | Ω | Ω ∂ t ̺ + div x ( ∇ x Φ + ̺ G W ) = 0 ∂ t V = 1 | Ω | div x ( ∇ x Φ + ̺ G W ) G W � � ∇ x M + ∇ x M ⊥ − 2 N div x M div x = div x ( ∇ x Φ + ̺ G W ) G W − ∂ t V

Additive noise, Step IV Step IV Fix h , H so that x M − 2 h = V + ∇ x Φ + ̺ G W , H = ∇ x M + ∇ t N div x M I ∈ R N × N 0 , sym Tranformed system III � ( v + h ) ⊗ ( v + h ) � ∂ t v + div x − H + p ( ̺ ) I + ∂ t Φ I = 0 ̺ div x v = 0 � v (0 , · ) = v 0 = H [ u 0 ] − 1 u 0 d x | Ω | Ω

Additive noise, Step V Prescribing the kinetic energy | v + h | 2 1 = e = Λ − N 2 ( p ( ̺ ) + ∂ t Φ) , Λ = Λ( t ) 2 ̺ Abstract Euler system � ( v + h ) ⊗ ( v + h ) � | v + h | 2 − 1 ∂ t v + div x I − H = 0 ̺ N ̺ | v + h | 2 div x v = 0 , 1 = e 2 ̺ v (0 , · ) = v 0 Random parameters The functions v 0 , h and H are random variables, the energy e can be taken deterministic.

Subsolutions Field equations, differential constraints ∂ t v + div x F = 0 , div x v = 0 v (0 , · ) = v 0 , v ( T , · ) = v T Non-linear constraint v ∈ C ([0 , T ] × Ω; R N ) , F ∈ C ([0 , T ] × Ω; R N × N sym , 0 ) , � ( v + h ) ⊗ ( v + h ) � N 2 λ max − F + M < e ̺

Subsolution relaxation Algebraic inequality � ( v + h ) ⊗ ( v + h ) � | v + h | 2 1 ≤ N − F + M 2 λ max < e 2 ̺ ̺ Solutions | v + h | 2 1 = e 2 ̺ ⇒ | v + h | 2 F = ( v + h ) ⊗ ( v + h ) − 1 I + M ̺ N ̺

Augmenting oscillations Oscillatory lemma If ̺, e , h ∈ C ( Q ; R N ) , ̺, e > 0 , H ∈ C ( Q ; R N × N sym , 0 ) � h ⊗ h � N 2 λ max − H < e in Q , ̺ then there exist w n ∈ C ∞ c ( Q ; R N ) , G n ∈ C ∞ c ( Q ; R N × N sym , 0 ) , n = 0 , 1 , . . . ∂ t w n + div x G n = 0 , div x w n = 0 in R × R N , � ( h + w n ) ⊗ ( h + w n ) � N 2 λ max − ( H + G n ) < e ̺ � � � � 2 | w n | 2 | h | 2 e − 1 w n ⇀ 0 , lim inf d x d t ≥ Λ(max Ω e ) d x d t 2 ̺ ̺ n →∞ Q Q

Basic ideas of proof [DeLellis and Sz´ ekelyhidi] Basic result Unit cube and constant coefficients ̺ , e , h , H Scaling Localizing the basic result to “small” cubes by means of scaling arguments Approximation Replacing all continuous functions by their means on any of the “small” cubes

Difficulties in the stochastic world Adaptiveness All quantities must be adapted to the filtration associated to the Wiener process W Geometric setting Continuous functions approximated in a similar way as in the definition of Itˆ o’s integral Admissible directions for oscillations selected by the Kuratowski, Ryll–Nardzewski theorem Space–time localization Stopping the Wiener process by its H¨ older norm

Stochastic version of the oscillatory lemma Fixing parameters Problem restricted to intervals small cubes [ t k , t k +1 ] × B k ( x ). All random parameters replaced by their values at t k Constructing oscillations Adapting the procedure by De Lellis and Sz´ ekelyhidi using Ryll–Nardzewski theorem on measurable selection Cutting off oscillatory increments The difference W ( t k ) − W ( t ) must remain small on [ t k , t k +1 ] - requires knowledge of the H¨ older constant of W on [ t k , t k +1 ] at t k - in general not predictable unless W is replaced by uniformly H¨ older function - the necessity of stopping times τ k .