On weak solution approach to problems in fluid dynamics Eduard Feireisl based on joint work with J.Březina (Tokio), C.Klingenberg, and S.Markfelder (Wuerzburg), O.Kreml (Praha), M. Lukáčová (Mainz), H.Mizerová (Bratislava), A.Novotný (Toulon) Institute of Mathematics, Academy of Sciences of the Czech Republic, Prague Technische Universität Berlin Hausdorff Kolloquium 2018, Bonn, 18 April, 2018
Why weak solutions? Everything Should Be Made as Simple as Possible, But Not Simpler... Albert Einstein [1874-1965] Weak solutions The largest possible class of objects that can be identified as solution of the a given problem, here a system of partial differential equations. The weaker the better Weak solutions are easy to be identified as asymptotic limits of approximate solutions, notably solutions of numerical schemes. A weak solution coincides with the (unique) strong solution as long as the latter exists. Weak strong uniqueness principle
Nonlinear balance laws Field equations tu + div xf(u) = G(U), U = [U 1,..., U M ], U = U(t, x) t (0, T ), x R N physical space: t the time variable, x the spatial variables Initial state U(0, ) = U 0 U, U 0 in state space F R M Admissibility conditions ts(u) + div xf S(U) = ( )G(U) U S(U) S - entropy
Approximate problems - numerics Approximate problem - discretization D tu n + div xf n(u n) = G n(u n) + error n Goal U n U in some sense, U exact solution Tools Make the target space for the limit as large as possible - easy convergence proof Show weak strong uniqueness
What can be weak... Weak derivatives - functions replaced by their spatial averages v = v(t, x) v(t, x)φ(t, x) dxdt Dv v(t, x)dφ(t, x) dxdt φ smooth test function 1 v(t, x) lim B(t,x) 0 B(t, x) B(t,x) v dxdt weak convergence = convergence in averages Measure valued (MV) solution v = v(t, x) V (t,x) ; v, {V} t,x family of probability measures F (v) V (t,x) ; F (v) for any Borel function F (MV) convergence = convergence in the weak topology of measures
Weak formulation of a general system Probability (Young) measure V t,x P(F) Field equations = T 0 [ R N V t,x; U ϕ ] t=τ R N V t,x; U tϕ + V t,x; F(U) xϕ V t,x; G(U) ϕdxdt Admissibility conditions [ T 0 t=0 ϕ smooth compactly supported in [0, T ) R N R N V t,x; S(U) ϕ ] t=τ R N V t,x; S(U) tϕ+ V t,x; F S(U) xϕ V t,x; G(U) U S(U) ϕdx dt t=0 =
Dynamics of compressible fluids Phase variables mass density ϱ = ϱ(t, x), t (0, T ), x R 3 (absolute) temperature ϑ = ϑ(t, x), t (0, T ), x R 3 (bulk) velocity field u = u(t, x), t (0, T ), x R 3 Standard formulation tϱ + div x(ϱu) = 0 t(ϱu) + div x(ϱu u) + xp(ϱ, ϑ) = 0 ( ) [( ) ] 1 1 t 2 ϱ u 2 + ϱe(ϱ, ϑ) + div x 2 ϱ u 2 + ϱe(ϱ, ϑ) + p(ϱ, ϑ) u = 0 Impermeability condition u n = 0
Complete Euler system in conservative variables Conservative variables Field equations mass density ϱ = ϱ(t, x), t (0, T ), x R 3 (total energy) E = E(t, x), t (0, T ), x R 3 momentum m = m(t, x), t (0, T ), x R 3 ( p = (γ 1)ϱe, p = (γ 1) E 1 2 ) m 2 ϱ tϱ + div xm = 0 tm + div x ( m m ϱ te + div x [ (E + p) m ϱ ) + xp = 0 ] = 0
Entropy Gibbs relation ϑds = De + pd ( ) 1 ϱ Entropy balance t(ϱs) + div x(sm) 0 Entropy in the polytropic case ( ) p s = S ϱ γ = S (γ 1) E 1 2 ϱ γ m 2 ϱ
Several concepts of solutions Classical solutions The phase variables are smooth (differentiable), the equations are satisfied in the standard sense. Classical solutions are often uniquely determined by the data. The main issue here is global in time existence that may fail for generic initial data Weak (distributional) solutions Limits of classical solutions, limits of regularized problems. Equations are satisfies in the distributional sense. Weak solutions may not be uniquely determined by the data. Viscosity solutions Limits of the Navier-Stokes-Fourier system for vanishing transport coefficients. Limits of approximate (numerical) schemes Zero step limits of numerical schemes. Examples are Lax Friedrichs and related schemes mimicking certain approximations - e.g. a model proposed by H.Brenner.
Admissible (entropy) weak solutions Field equations [ [ ϱϕ dx m ϕ dx [ Eϕ dx ] t=τ t=0 ] t=τ t=0 ] t=τ t=0 = τ 0 [ϱ tϕ + ϱu xϕ] dxdt for all ϕ C 1 ([0, T ] ) τ [ = m tϕ + m m ] : xϕ + pdiv xϕ ϱ 0 for all ϕ C 1 ([0, T ] ; R 3 ), ϕ n = 0 τ [ [ = E tϕ + (E + p) m ] ] xϕ dxdt ϱ 0 for all ϕ C 1 ([0, T ] ) dxdt Entropy inequality [ ϱsϕ dx ] t=τ t=0 τ 0 [ϱs tϕ + sm xϕ] dxdt for all ϕ C 1 ([0, T ] ), ϕ 0
Infinitely many weak solutions Initial data ϱ(0, ) = ϱ 0, u(0, ) = u 0, ϑ(0, ) = ϑ 0. Existence via convex integration Let N = 2, 3. Let ϱ 0, ϑ 0 be piecewise constant (arbitrary) positive. Then the exists u 0 L such that the Euler system admits infinitely many admissible weak solution in (0, T ).
Dissipative measure valued (DMV) solutions Parameterized measure { } }{{} F phase space = ϱ 0, m R 3, E [0, ), Q }{{} T physical space = (0, T ) {V t,x} (t,x) QT, Y t,x P(F) Field equations t V t,x; ϱ + div x V t,x; m = 0 t V t,x; m + div x V m m t,x; + x V t,x; p = D xµ C ϱ t V t,x; E dx + D = 0, t V t,x; ϱs + div x V t,x; sm 0 Compatibility τ 0 µ C dxdt C τ 0 Ddt
Why to go measure valued? Motto: The larger (class) the better Universal limits of numerical schemes Limits of more complex physical systems - vanishing viscosity/heat conductivity limit Singular limits (low Mach etc.) Weak strong uniqueness A (DMV) solution coincides with a smooth solution with the same initial data as long as the latter solution exists
Thermodynamic stability Thermodynamic stability in the standard variables p(ϱ, ϑ) ϱ > 0, e(ϱ, ϑ) ϑ > 0 Thermodynamic stability in the conservative variables is a (strictly) concave function (ϱ, m, E) ϱs(ϱ, m, E) Thermodynamic stability in the polytropic case ( ) p ϱs = ϱs, p = (γ 1)ϱe ϱ γ S (Z) > 0, (1 γ)s (Z) γs (Z)Z > 0
Relative energy Relative energy in the standard variables ( E ϱ, ϑ, u ϱ, ϑ, ) ũ = 1 2 ϱ u ũ 2 + ϱh ϑ( ϱ, ϑ)(ϱ ϱ) H ϑ( ϱ, ϑ) ( H ϑ(ϱ, ϑ) = ϱ e(ϱ, ϑ) ϑs(ϱ, ) ϑ) Relative energy in the conservative variables ( ) E ϱ, m, E ϱ, m, Ẽ = ϑ [ ϱs ϱ(ϱs)(ϱ ϱ) m(ϱs) (m m) E (ϱs)(e Ẽ) ] ϱ s
Relative energy inequality Relative energy revisited ( E ϱ, m, E ϱ, ϑ, ) ũ E ϑs(ϱ, m, E) m ũ + 1 2 ϱ ũ 2 + p( ϱ, ϑ) ( ) p( ϱ, ϑ) e( ϱ, ϑ) ϑs( ϱ, ϑ) + ϱ ϱ Relative energy inequality [ ( ) V t,x; E ϱ, m, E ϱ, ϑ, ũ ] t=τ τ dx + D(τ) R(t) dt t=0 0
Stability of strong solutions Measure valued strong uniqueness Suppose the thermodynamic functions p, e, and s comply with the hypothesis of thermodynamic stability. Let (ϱ, m, E) be a smooth (C 1 ) solution of the Euler system and let (Y t,x; D) be a dissipative measure valued solution of the same system with the same initial data, meaning Y 0,x = δ ϱ0 (x),m 0 (x),e 0 (x) for a.a. x. Then for a.a. (t, x) (0, T ). D 0, Y t,x = δ ϱ(t,x),m(t,x),e(t,x)
Maximal dissipation principle Entropy production rate t(ϱs) + div x(ϱm) = σ 0 Dissipative ordering T 0 T 0 Vt,x 1 Vt,x 2 iff σ 1 σ 2 in [0, T ) [ Vt,x; 1 S(ϱ, m, E) tϕ + Vt,x; 1 S(ϱ, m, E) m ] xϕ ϱ [ Vt,x; 2 S(ϱ, m, E) tϕ + Vt,x; 2 S(ϱ, m, E) m ] xϕ ϱ Maximal dissipation principle A (DMV) solution is admissible if it is maximal with respect to the ordering. A maximal (DMV) solution exists. dxdt dxdt
Generating MV solutions - zero viscosity limit Navier Stokes Fourier system tϱ + div x(ϱu) = 0 t(ϱu) + div x (ϱu u) + xp = εdiv xs t (ϱe) + div x (ϱeu) + ε xq = εs : xu pdiv xu Physical dissipation S(ϑ, xu) = µ(ϑ) ( xu + txu 23 divxui ), q = κ(ϑ) xϑ
Generating MV solutions - artificial viscosity Lax Friedrichs numerical scheme tϱ + div x(ϱu) = εdiv x(λ xϱ) t(ϱu) + div x (ϱu u) + xp = εdiv x(λ x(ϱu)) te + div x((e + p)u) = εdiv x(λ xe) Entropy preserving (ϱs) + div x(ϱsu) εdiv x(λϱ xs)) + defect
Brenner s model Two velocities principle u u m = εk x log(ϱ) Field equations tϱ + div x(ϱu m) = 0 t(ϱu) + div x (ϱu u m) + xp(ϱ, ϑ) = εdiv xs t (ϱ( 1 ) 2 u 2 + e(ϱ, ϑ)) + div x (ϱ( 12 ) u 2 + e(ϱ, ϑ))u m + div x(p(ϱ, ϑ)u) + εdiv xq = εdiv x(su) Constitutive relations S = µ ( xu + txu 23 ) divxu, q = κ xϑ, K = κ c v ϱ