Weak formulation of a nonlinear PDE system arising from models of phase transitions and damage

Transcription

1 Weak formulation of a nonlinear PDE system arising from models of phase transitions and damage E. Rocca Università degli Studi di Milano joint work with Riccarda Rossi (Università di Brescia, Italy) Supported by the FP7-IDEAS-ERC-StG Grant EntroPhase # E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

2 Outline E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

3 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

4 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation Handle nonlinearities and degeneracy in the momentum balance + order parameter equation = generalization of the principle of virtual powers applied to E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

5 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation Handle nonlinearities and degeneracy in the momentum balance + order parameter equation = generalization of the principle of virtual powers applied to the non degenerating case [joint works with R. Rossi, J. Differential Equations and Appl. Math (2008)] the degenerate case [joint work with R. Rossi, preprint arxiv: v1 (2012)] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

6 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation Handle nonlinearities and degeneracy in the momentum balance + order parameter equation = generalization of the principle of virtual powers applied to the non degenerating case [joint works with R. Rossi, J. Differential Equations and Appl. Math (2008)] the degenerate case [joint work with R. Rossi, preprint arxiv: v1 (2012)] Handle the high order nonlinearities in the internal energy balance = entropic formulation applied to E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

7 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation Handle nonlinearities and degeneracy in the momentum balance + order parameter equation = generalization of the principle of virtual powers applied to the non degenerating case [joint works with R. Rossi, J. Differential Equations and Appl. Math (2008)] the degenerate case [joint work with R. Rossi, preprint arxiv: v1 (2012)] Handle the high order nonlinearities in the internal energy balance = entropic formulation applied to a solid-liquid phase transition model [joint work with E. Feireisl, H. Petzeltová, M2AS (2009)] the damage phenomena [work in progress with R. Rossi] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

8 Outline Introduce the nonlinear PDE system coupling - an internal energy balance - a (possibly degenerating) momentum balance - an order parameter equation Handle nonlinearities and degeneracy in the momentum balance + order parameter equation = generalization of the principle of virtual powers applied to the non degenerating case [joint works with R. Rossi, J. Differential Equations and Appl. Math (2008)] the degenerate case [joint work with R. Rossi, preprint arxiv: v1 (2012)] Handle the high order nonlinearities in the internal energy balance = entropic formulation applied to a solid-liquid phase transition model [joint work with E. Feireisl, H. Petzeltová, M2AS (2009)] the damage phenomena [work in progress with R. Rossi] Present other possible applications of these formulations to: phase separation, liquid crystals, immiscible fluids E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

9 Aim: deal with diffuse interface models in thermoviscoelasticity: phase transitions in thermoviscoelastic materials and non-isothermal models for damage phenomena E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

10 Aim: deal with diffuse interface models in thermoviscoelasticity: phase transitions in thermoviscoelastic materials and non-isothermal models for damage phenomena Concentrate on damage phenomena accounting for - the absolute temperature θ - the evolution of the (small) displacement variables u (ε ij (u) := (u i,j + u j,i )/2, i, j = 1, 2, 3) - the damage parameter χ [0, 1]: χ = 0 (completely damaged), χ = 1 (completely undamaged) where the internal energy balance displays nonlinear dissipation and the momentum equation contains χ-dependent elliptic operators, degenerating at the pure phase χ = 0 c(θ)θ t + χ tθ + ρθ div u t div(k(θ) θ)) = g + χ ε(u t) 2 + χ t 2 u tt div(χε(u t) + χε(u) ρθ1) = f χ t + I (,0] (χ t) + A sχ + W (χ) ε(u) θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

11 Aim: deal with diffuse interface models in thermoviscoelasticity: phase transitions in thermoviscoelastic materials and non-isothermal models for damage phenomena Concentrate on damage phenomena accounting for - the absolute temperature θ - the evolution of the (small) displacement variables u (ε ij (u) := (u i,j + u j,i )/2, i, j = 1, 2, 3) - the damage parameter χ [0, 1]: χ = 0 (completely damaged), χ = 1 (completely undamaged) where the internal energy balance displays nonlinear dissipation and the momentum equation contains χ-dependent elliptic operators, degenerating at the pure phase χ = 0 c(θ)θ t + χ tθ + ρθ div u t div(k(θ) θ)) = g + χ ε(u t) 2 + χ t 2 u tt div(χε(u t) + χε(u) ρθ1) = f χ t + I (,0] (χ t) + A sχ + W (χ) ε(u) θ Unidirectional: I (,0] (χ t) = 0 if χ t (, 0], I (,0] (χ t) = + otherwise Nonlocal: A s : H s () H s () the fractional s Laplacian (s > d/2) W = β + γ, γ C 2 (R), β proper, convex, l.s.c., dom( β) = [0, 1] (e.g. β = I[0,1] ) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

12 The main ideas to handle nonlinearities and degeneracy E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

13 The main ideas to handle nonlinearities and degeneracy Combining the concept of weak solution satisfying: E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

14 The main ideas to handle nonlinearities and degeneracy Combining the concept of weak solution satisfying: 1. a suitable energy conservation and entropy inequality inspired by: 1.1. the works of E. Feireisl and co-authors ([Feireisl, Comput. Math. Appl. (2007)] and [Buĺıček, Feireisl, & Málek, Nonlinear Anal. Real World Appl. (2009)]) for heat conduction in fluids = weak formulation of the internal energy balance E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

15 The main ideas to handle nonlinearities and degeneracy Combining the concept of weak solution satisfying: 1. a suitable energy conservation and entropy inequality inspired by: 1.1. the works of E. Feireisl and co-authors ([Feireisl, Comput. Math. Appl. (2007)] and [Buĺıček, Feireisl, & Málek, Nonlinear Anal. Real World Appl. (2009)]) for heat conduction in fluids = weak formulation of the internal energy balance 2. a generalization of the principle of virtual powers inspired by: 2.1. a notion of weak solution introduced by [Heinemann, Kraus, WIAS preprint 1569 and Adv. Math. Sci. Appl. (2011)] for non-degenerating isothermal diffuse interface models for phase separation and damage = weak formulation of the phase equation 2.2. the notion of energetic solution - A. Mielke and co-authors ([Bouchitté, Mielke, Roubíček, ZAMP. Angew. Math. Phys. (2009) and [Mielke, Roubíček, Zeman, Comput. Methods Appl. Mech. Engrg. (2010)]) for rate-independent processes for complete damage = weak formulation of the momentum balance E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

16 Weak formulation of the χ+u-equations: the generalized principle of virtual powers E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

17 The first aim: to concentrate on degeneracy and nonlinearities in the u+χ-equations E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

18 The first aim: to concentrate on degeneracy and nonlinearities in the u+χ-equations [E.R., R. Rossi, preprint arxiv: v1 (2012)] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

19 The first aim: to concentrate on degeneracy and nonlinearities in the u+χ-equations [E.R., R. Rossi, preprint arxiv: v1 (2012)] neglect the nonlinear terms χ ε(u t) 2 + χ t 2 on the r.h.s (using the small perturbations assumption) in the internal energy balance c(θ)θ t + χ tθ + ρθ div u t div(k(θ) θ) = g +χε(u t) 2 + χ t 2 }{{} = 0 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

20 The first aim: to concentrate on degeneracy and nonlinearities in the u+χ-equations [E.R., R. Rossi, preprint arxiv: v1 (2012)] neglect the nonlinear terms χ ε(u t) 2 + χ t 2 on the r.h.s (using the small perturbations assumption) in the internal energy balance c(θ)θ t + χ tθ + ρθ div u t div(k(θ) θ) = g +χε(u t) 2 + χ t 2 }{{} = 0 give a weak formulation of the (degenerate) momentum balance and the phase equation (principle of virtual powers) u tt div(χε(u t) + χε(u) ρθ1) = f χ t + I (,0] (χ t)+a sχ + W (χ) ε(u) 2 2 in order to handle the degeneracy and the nonlinearities in the u+χ-equations. + θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

21 The non-degenerating approximation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

22 The non-degenerating approximation Main aim: We shall let χ vanishes at the threshold value 0, not enforce separation of χ from the threshold value 0, and accordingly we will allow for general initial configurations of χ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

23 The non-degenerating approximation Main aim: We shall let χ vanishes at the threshold value 0, not enforce separation of χ from the threshold value 0, and accordingly we will allow for general initial configurations of χ = We shall approximate the system with a non-degenerating one, where we replace the momentum equation with u tt div((χ + δ)ε(u t) + (χ + δ)ε(u) ρθ1) = f for δ > 0 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

24 The non-degenerating approximation Main aim: We shall let χ vanishes at the threshold value 0, not enforce separation of χ from the threshold value 0, and accordingly we will allow for general initial configurations of χ = We shall approximate the system with a non-degenerating one, where we replace the momentum equation with u tt div((χ + δ)ε(u t) + (χ + δ)ε(u) ρθ1) = f for δ > 0 Note that both the coefficients need to be truncated when taking the degenerate limit in the momentum equation: the truncation in front of ε(u t) allows us to deal with the main part of the elliptic operator but in order to pass to the limit in the quadratic term on the right-hand side of χ-eq., we also need to truncate the coefficient of ε(u) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

25 Our results E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

26 Our results [First result] Local in time well-posedness for a suitable formulation of the reversible problem, using in θ t + χ tθ θ = g +χε(u t) 2 + χ t 2 }{{} = 0 the small perturbations assumption in the 3D (in space) setting [E.R., Rossi, J. Differential Equations, 2008] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

27 Our results [First result] Local in time well-posedness for a suitable formulation of the reversible problem, using in θ t + χ tθ θ = g +χε(u t) 2 + χ t 2 }{{} = 0 the small perturbations assumption in the 3D (in space) setting [E.R., Rossi, J. Differential Equations, 2008] [Second result] Global well-posedness in the 1D case without small perturbations assumption [E.R, Rossi, Appl. Math., Special Volume (2008)] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

28 Our results [First result] Local in time well-posedness for a suitable formulation of the reversible problem, using in θ t + χ tθ θ = g +χε(u t) 2 + χ t 2 }{{} = 0 the small perturbations assumption in the 3D (in space) setting [E.R., Rossi, J. Differential Equations, 2008] [Second result] Global well-posedness in the 1D case without small perturbations assumption [E.R, Rossi, Appl. Math., Special Volume (2008)] Note: in both these results we assumed χ 0 separated from the thresholds 0 and 1 and we prove (via coercivity condition on W at the thresholds 0 and 1) that the solution χ of χ t χ + W (χ) ε(u) 2 2 during the evolution continues to stay separated from 0 and 1 = prevent degeneracy (the operators are uniformly elliptic) + θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

29 Our results [First result] Local in time well-posedness for a suitable formulation of the reversible problem, using in θ t + χ tθ θ = g +χε(u t) 2 + χ t 2 }{{} = 0 the small perturbations assumption in the 3D (in space) setting [E.R., Rossi, J. Differential Equations, 2008] [Second result] Global well-posedness in the 1D case without small perturbations assumption [E.R, Rossi, Appl. Math., Special Volume (2008)] Note: in both these results we assumed χ 0 separated from the thresholds 0 and 1 and we prove (via coercivity condition on W at the thresholds 0 and 1) that the solution χ of χ t χ + W (χ) ε(u) 2 2 during the evolution continues to stay separated from 0 and 1 = prevent degeneracy (the operators are uniformly elliptic) [The last result] [E.R., R. Rossi, preprint arxiv: v1 (2012)]: global existence result in 3D using a suitable notion of solution and without enforcing the separation property, i.e. allowing for degeneracy = we need a s-laplacian or a p-laplacian on χ + θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

30 The free-energy cf. [M. Frémond, Phase Change in Mechanics, Lecture Notes of UMI, Springer-Verlag, 2012] F = (f (θ) + χ ε(u) 2 as(χ, χ) ) + + W (χ) θχ dx 2 2 the stiffness of the material decreases as χ 0 f is a concave function, the heat capacity is c(θ) = θf (θ) ( z1 (x) z 1 (y) ) ( z 2 (x) z 2 (y) ) a s(z 1, z 2 ) := x y d+2(s 1) dx dy is the bilinear form associated to the nonlocal fractional s Laplacian A s (or a p(χ, χ) = χ p /p) s > d/2 (or p > d): we need the compact embedding of H s () into C 0 () W = β + γ, γ C 2 (R), β proper, convex, l.s.c., dom( β) = [0, 1] (e.g. β = I[0,1] ) we can include the thermal expansion term ρθtr(ε(u)) (neglected in this presentation) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

31 The free-energy cf. [M. Frémond, Phase Change in Mechanics, Lecture Notes of UMI, Springer-Verlag, 2012] F = (f (θ) + χ ε(u) 2 as(χ, χ) ) + + W (χ) θχ dx 2 2 the stiffness of the material decreases as χ 0 f is a concave function, the heat capacity is c(θ) = θf (θ) ( z1 (x) z 1 (y) ) ( z 2 (x) z 2 (y) ) a s(z 1, z 2 ) := x y d+2(s 1) dx dy is the bilinear form associated to the nonlocal fractional s Laplacian A s (or a p(χ, χ) = χ p /p) s > d/2 (or p > d): we need the compact embedding of H s () into C 0 () W = β + γ, γ C 2 (R), β proper, convex, l.s.c., dom( β) = [0, 1] (e.g. β = I[0,1] ) we can include the thermal expansion term ρθtr(ε(u)) (neglected in this presentation) The pseudo-potential P = k(θ) 2 θ χt 2 + χ ε(ut) 2 + I (,0] (χ t) 2 k is the heat conductivity I (,0] (χ t) = 0 if χ t (, 0], I (,0] (χ t) = + otherwise (irreversibility of the damage) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

32 The modelling The momentum equation u tt div σ = f ( σ = σ d + σ nd = P ε(u t ) + u tt div(χε(u t) + χε(u)) = f ) F ε(u) becomes E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

33 The modelling The momentum equation u tt div σ = f ( σ = σ d + σ nd = P ε(u t ) + u tt div(χε(u t) + χε(u)) = f ) F ε(u) becomes The standard principle of virtual powers ( ) B div H = 0 B = P + F χ t χ, H = F χ becomes χ t + I (,0] (χ t)+a sχ + W (χ) ε(u) θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

34 The modelling The momentum equation u tt div σ = f ( σ = σ d + σ nd = P ε(u t ) + u tt div(χε(u t) + χε(u)) = f ) F ε(u) becomes The standard principle of virtual powers ( ) B div H = 0 B = P + F χ t χ, H = F χ becomes χ t + I (,0] (χ t)+a sχ + W (χ) ε(u) θ The internal energy balance becomes e t + div q = g + σ : ε(u t) + Bχ t + H χ t ( e = F θ F θ, ) q = P θ c(θ)θ t + χ tθ div(k(θ) θ) = g+χ ε(u t) 2 + χ t 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

35 The technique E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

36 The technique We replace the momentum equation with a non-degenerating one u tt div((χ + δ)ε(u t) + (χ + δ)ε(u)) = f, δ > 0 (1) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

37 The technique We replace the momentum equation with a non-degenerating one u tt div((χ + δ)ε(u t) + (χ + δ)ε(u)) = f, δ > 0 (1) We have to handle the nonlinear coupling between the single equations: in the heat equation (even using the small perturbation assumption) and in the phase equation c(θ)θ t + χ tθ div(k(θ) θ) = g χ t + I (,0] (χ t) + A sχ + W (χ) ε(u) θ (2) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

38 The technique We replace the momentum equation with a non-degenerating one u tt div((χ + δ)ε(u t) + (χ + δ)ε(u)) = f, δ > 0 (1) We have to handle the nonlinear coupling between the single equations: in the heat equation (even using the small perturbation assumption) and in the phase equation c(θ)θ t + χ tθ div(k(θ) θ) = g χ t + I (,0] (χ t) + A sχ + W (χ) ε(u) θ (2) A major difficulty stems from the simultaneous presence in (2) of I (,0] (χ t) and W (χ) and from the low regularities of ε(u) θ on the r.h.s. = follow the approach of [Heinemann, Kraus, Adv. Math. Sci. Appl. (2011)] and consider a suitable weak formulation of (2) consisting of a one-sided variational inequality + an energy inequality = generalized principle of virtual powers E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

39 The technique We replace the momentum equation with a non-degenerating one u tt div((χ + δ)ε(u t) + (χ + δ)ε(u)) = f, δ > 0 (1) We have to handle the nonlinear coupling between the single equations: in the heat equation (even using the small perturbation assumption) and in the phase equation c(θ)θ t + χ tθ div(k(θ) θ) = g χ t + I (,0] (χ t) + A sχ + W (χ) ε(u) θ (2) A major difficulty stems from the simultaneous presence in (2) of I (,0] (χ t) and W (χ) and from the low regularities of ε(u) θ on the r.h.s. = follow the approach of [Heinemann, Kraus, Adv. Math. Sci. Appl. (2011)] and consider a suitable weak formulation of (2) consisting of a one-sided variational inequality + an energy inequality = generalized principle of virtual powers For the analysis of the degenerate limit δ 0 we have carefully adapted techniques from [Bouchitté, Mielke, Roubíček, ZAMP (2009)] and [Mielke, Roubíček, Zeman, Comput. Methods Appl. Mech. Engrg. (2011)] to the case of a rate-dependent equation for χ, also coupled with the temperature equation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

40 Energy vs Enthalpy In order to deal with the low regularity of θ, rewrite the internal energy equation as the enthalpy equation c(θ)θ t + χ tθ div(k(θ) θ) = g w t + χ tθ(w) div(k(w) w) = g where { θ h 1 (w) if w 0, w = h(θ) := c(s) ds, Θ(w) := K(w) := k(θ(w)) 0 0 if w < 0, c(θ(w)) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

41 Energy vs Enthalpy In order to deal with the low regularity of θ, rewrite the internal energy equation as the enthalpy equation c(θ)θ t + χ tθ div(k(θ) θ) = g w t + χ tθ(w) div(k(w) w) = g where { θ h 1 (w) if w 0, w = h(θ) := c(s) ds, Θ(w) := K(w) := k(θ(w)) 0 0 if w < 0, c(θ(w)) We assume that c C 0 ([0, + ); [0, + )) σ 1 σ > 2d d+2 : c0(1+θ)σ 1 c(θ) c 1(1+θ) σ 1 1 = h is strictly increasing the function k : [0, + ) [0, + ) is continuous, and c 2, c 3 > 0 θ [0, + ) : c 2c(θ) k(θ) c 3(c(θ) + 1) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

42 Energy vs Enthalpy In order to deal with the low regularity of θ, rewrite the internal energy equation as the enthalpy equation c(θ)θ t + χ tθ div(k(θ) θ) = g w t + χ tθ(w) div(k(w) w) = g where { θ h 1 (w) if w 0, w = h(θ) := c(s) ds, Θ(w) := K(w) := k(θ(w)) 0 0 if w < 0, c(θ(w)) We assume that c C 0 ([0, + ); [0, + )) σ 1 σ > 2d d+2 : c0(1+θ)σ 1 c(θ) c 1(1+θ) σ 1 1 = h is strictly increasing the function k : [0, + ) [0, + ) is continuous, and c 2, c 3 > 0 θ [0, + ) : c 2c(θ) k(θ) c 3(c(θ) + 1) = c > 0 w R : c 2 K(w) c = for every s (1, ) C s > 0 w L 1 () : Θ(w) L s () C s( w 1/σ L s/σ () + 1) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

43 The approximating non-degenerate Problem [P δ ] Given δ > 0, take W = I [0,+ ) + γ, γ C 1 (R), find (measurable) functions w L r (0, T ; W 1,r ()) L (0, T ; L 1 ()) BV([0, T ]; W 1,r () ) u H 1 (0, T ; H 2 (; R d )) W 1, (0, T ; H 1 0 ()) H2 (0, T ; L 2 (; R d )) χ L (0, T ; H s ()) H 1 (0, T ; L 2 ()) for every 1 r < d+2, fulfilling the initial conditions d+1 u(0, x) = u 0 (x), u t(0, x) = v 0 (x) for a.e. x χ(0, x) = χ 0 (x) for a.e. x the equations (for every ϕ C 0 ([0, T ]; W 1,r ()) W 1,r (0, T ; L r ()) and t (0, T ]) t t t ϕ(t) w(t)(dx) wϕ t dx + χ tθ(w)ϕ dx + K(w) w ϕ dx t = gϕ + w 0 ϕ(0) dx 0 u tt div ((χ + δ)ε(u t) + (χ + δ)ε(u)) = f in H 1 (; R d ) a.e. in (0, T ) and the subdifferential inclusion in a suitable sense χ t + I (,0] (χ t) + A sχ + I [0,+ ) (χ) + γ(χ) ε(u) Θ(w) in H s () and a.e. in (0, T ) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

44 Generalized principle of virtual powers for δ > 0 [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 1] (δ > 0) Under the previous assumptions on the data (W = I [0,+ ) + γ), then, [1.] Problem [P δ ] admits a weak solution (w, u, χ), which, beside fulfilling the enthalpy and momentum equations, satisfies χ t(x, t) 0 for almost all t (0, T ), and ( ϕ L 2 (0, T ; H+ s ()) L (Q)) the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ with ξ I [0,+ ) (χ) in the following sense: ξ L 1 (0, T ; L 1 ()), ξ(t), ϕ χ(t) H s () 0 ϕ Hs + (), a.e. t (0, T ) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

45 Generalized principle of virtual powers for δ > 0 [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 1] (δ > 0) Under the previous assumptions on the data (W = I [0,+ ) + γ), then, [1.] Problem [P δ ] admits a weak solution (w, u, χ), which, beside fulfilling the enthalpy and momentum equations, satisfies χ t(x, t) 0 for almost all t (0, T ), and ( ϕ L 2 (0, T ; H+ s ()) L (Q)) the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ with ξ I [0,+ ) (χ) in the following sense: ξ L 1 (0, T ; L 1 ()), ξ(t), ϕ χ(t) H s () 0 ϕ Hs + (), a.e. t (0, T ) and the energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t: t χ t 2 dx dr + 1 τ 2 as(χ(t), χ(t)) + W (χ(t)) dx 1 t ( ) 2 as(χ(τ), χ(τ)) + W (χ(τ)) dx + χ t ε(u) 2 + Θ(w) dx dr τ 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

46 Generalized principle of virtual powers for δ > 0 [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 1] (δ > 0) Under the previous assumptions on the data (W = I [0,+ ) + γ), then, [1.] Problem [P δ ] admits a weak solution (w, u, χ), which, beside fulfilling the enthalpy and momentum equations, satisfies χ t(x, t) 0 for almost all t (0, T ), and ( ϕ L 2 (0, T ; H+ s ()) L (Q)) the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ with ξ I [0,+ ) (χ) in the following sense: ξ L 1 (0, T ; L 1 ()), ξ(t), ϕ χ(t) H s () 0 ϕ Hs + (), a.e. t (0, T ) and the energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t: t χ t 2 dx dr + 1 τ 2 as(χ(t), χ(t)) + W (χ(t)) dx 1 t ( ) 2 as(χ(τ), χ(τ)) + W (χ(τ)) dx + χ t ε(u) 2 + Θ(w) dx dr τ 2 [2.] Suppose in addition that g(x, t) 0, θ 0 > θ 0 0 a.e. Then θ(x, t) := Θ(w(x, t)) θ 0 0 a.e. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

47 Generalized principle of virtual powers for δ > 0 [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 1] (δ > 0) Under the previous assumptions on the data (W = I [0,+ ) + γ), then, [1.] Problem [P δ ] admits a weak solution (w, u, χ), which, beside fulfilling the enthalpy and momentum equations, satisfies χ t(x, t) 0 for almost all t (0, T ), and ( ϕ L 2 (0, T ; H+ s ()) L (Q)) the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ with ξ I [0,+ ) (χ) in the following sense: ξ L 1 (0, T ; L 1 ()), ξ(t), ϕ χ(t) H s () 0 ϕ Hs + (), a.e. t (0, T ) and the energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t: t χ t 2 dx dr + 1 τ 2 as(χ(t), χ(t)) + W (χ(t)) dx 1 t ( ) 2 as(χ(τ), χ(τ)) + W (χ(τ)) dx + χ t ε(u) 2 + Θ(w) dx dr τ 2 [2.] Suppose in addition that g(x, t) 0, θ 0 > θ 0 0 a.e. Then θ(x, t) := Θ(w(x, t)) θ 0 0 a.e. Uniqueness of solutions for the irreversible system, even in the isothermal case, is still an open problem. This is mainly due to the doubly nonlinear character of the χ equation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

48 Generalized principle of virtual powers vs classical phase inclusion E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

49 Generalized principle of virtual powers vs classical phase inclusion If (w, u, χ) are more regular and satisfy the notion of weak solution: the one-sided inequality ( ϕ L 2 (0, T ; H s ()) L (Q)): T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ with ξ I [0,+ ) (χ) and the energy inequality: t χ t 2 dx dr + 1 τ 2 as(χ(t), χ(t)) + W (χ(t)) dx 1 t ( ) 2 as(χ(τ), χ(τ)) + W (χ(τ)) dx + χ t ε(u) 2 + Θ(w) dx dr τ 2 Differentiating in time the energy inequality (energy) and using the chain rule, we conclude that (w, u, χ, ξ) comply with (one-sided) (energy) χ t(t) + A s(χ(t)) + ξ(t) + γ(χ(t)) + ε(u) 2 2 Θ(w(t)), χ t(t) 0 for a.e.t H s () (one-sided) (ineq) + χ t 0 a.e. are equivalent to the usual phase inclusion (ineq) χ t + A sχ + ξ + γ(χ) + ε(u) 2 2 Θ(w) I (,0] (χ t) in H s () E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

50 The techniques used in the proof of Thm. 1 (δ > 0) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

51 The techniques used in the proof of Thm. 1 (δ > 0) We pass to the limit in a carefully designed time-discretization scheme E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

52 The techniques used in the proof of Thm. 1 (δ > 0) We pass to the limit in a carefully designed time-discretization scheme Any weak solution (w, u, χ) fulfills the total energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t w(t)(dx) + 1 t t u t(t) 2 dx + χ t 2 dx dr + (χ + δ) ε(u t) 2 dx dr 2 τ τ + 1 (χ + δ) ε(u(t)) 2 dx + a s(χ(t), χ(t)) + W (χ(t)) dx 2 w(τ)(dx) + 1 u t(τ) 2 dx + 1 (χ + δ) ε(u(τ)) 2 dx + a s(χ(τ), χ(τ)) 2 2 t t + W (χ(τ)) dx + f u t dxdr + g dxdr τ τ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

53 The techniques used in the proof of Thm. 1 (δ > 0) We pass to the limit in a carefully designed time-discretization scheme Any weak solution (w, u, χ) fulfills the total energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t w(t)(dx) + 1 t t u t(t) 2 dx + χ t 2 dx dr + (χ + δ) ε(u t) 2 dx dr 2 τ τ + 1 (χ + δ) ε(u(t)) 2 dx + a s(χ(t), χ(t)) + W (χ(t)) dx 2 w(τ)(dx) + 1 u t(τ) 2 dx + 1 (χ + δ) ε(u(τ)) 2 dx + a s(χ(τ), χ(τ)) 2 2 t t + W (χ(τ)) dx + f u t dxdr + g dxdr τ The presence of the s-laplacian with s > d/2 = an estimate for χ in L (0, T ; H s ()) (from the total energy balance) = we can now test the momentum balance by div(ε(u t)) = an L (0, T ; L 2 ())-bound on the quadratic nonlinearity ε(u) 2 on the right-hand side of χ t + I (,0] (χ t)+a sχ + W (χ) ε(u) 2 2 τ + Θ(w) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

54 The techniques used in the proof of Thm. 1 (δ > 0) We pass to the limit in a carefully designed time-discretization scheme Any weak solution (w, u, χ) fulfills the total energy inequality for all t (0, T ], for τ = 0, and for almost all 0 < τ t w(t)(dx) + 1 t t u t(t) 2 dx + χ t 2 dx dr + (χ + δ) ε(u t) 2 dx dr 2 τ τ + 1 (χ + δ) ε(u(t)) 2 dx + a s(χ(t), χ(t)) + W (χ(t)) dx 2 w(τ)(dx) + 1 u t(τ) 2 dx + 1 (χ + δ) ε(u(τ)) 2 dx + a s(χ(τ), χ(τ)) 2 2 t t + W (χ(τ)) dx + f u t dxdr + g dxdr τ The presence of the s-laplacian with s > d/2 = an estimate for χ in L (0, T ; H s ()) (from the total energy balance) = we can now test the momentum balance by div(ε(u t)) = an L (0, T ; L 2 ())-bound on the quadratic nonlinearity ε(u) 2 on the right-hand side of χ t + I (,0] (χ t)+a sχ + W (χ) ε(u) 2 2 τ + Θ(w) A Boccardo-Gallouët-type estimate + Gagliardo-Nirenberg inequality lead to an L r (0, T ; W 1,r ())-estimate on the enthalpy w (and hence on Θ(w)) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

55 The total energy inequality in the degenerating case δ 0 Rewrite the momentum equation 2 t u δ div((χ + δ)ε( tu δ )) div((χ + δ)ε(u δ )) = f using the new variables (quasi-stresses) µ δ := χ δ + δ ε( tu δ ), and η δ := χ δ + δ ε(u δ ): 2 t u δ div( χ + δ µ δ ) div( χ + δ η δ ) = f E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

56 The total energy inequality in the degenerating case δ 0 Rewrite the momentum equation 2 t u δ div((χ + δ)ε( tu δ )) div((χ + δ)ε(u δ )) = f using the new variables (quasi-stresses) µ δ := χ δ + δ ε( tu δ ), and η δ := χ δ + δ ε(u δ ): 2 t u δ div( χ + δ µ δ ) div( χ + δ η δ ) = f The total energy inequality for (w δ, u δ, χ δ ) is w δ (t)(dx) + 1 t tu δ (t) 2 dx + tχ δ 2 dx + 1 t µ 2 τ 2 δ 2 dx τ + η δ (t) as(χ δ(t), χ δ (t)) + W (χ δ (t)) dx w δ (τ)(dx) + 1 tu δ (τ) 2 dx + η δ (τ) as(χ δ(τ), χ δ (τ)) t t + W (χ δ (τ)) dx + f tu δ dx + g dx τ τ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

57 The degenerate problem (δ = 0): the existence theorem [E.R., R. Rossi, preprint arxiv: v1 (2012)] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

58 The degenerate problem (δ = 0): the existence theorem [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 2] (δ = 0) Under the previous assumptions, there exist u W 1, (0, T ; L 2 ()) H 2 (0, T ; H 1 ()), µ L 2 (0, T ; L 2 ()), η L (0, T ; L 2 ()), w L r (0, T ; W 1,r ()) L (0, T ; L 1 ()) BV([0, T ]; W 1,r () ) χ L (0, T ; H s ()) H 1 (0, T ; L 2 ()), χ(x, t) 0, χ t(x, t) 0 a.e. such that E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

59 The degenerate problem (δ = 0): the existence theorem [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 2] (δ = 0) Under the previous assumptions, there exist u W 1, (0, T ; L 2 ()) H 2 (0, T ; H 1 ()), µ L 2 (0, T ; L 2 ()), η L (0, T ; L 2 ()), w L r (0, T ; W 1,r ()) L (0, T ; L 1 ()) BV([0, T ]; W 1,r () ) χ L (0, T ; H s ()) H 1 (0, T ; L 2 ()), χ(x, t) 0, χ t(x, t) 0 a.e. such that it holds true (a.e. in any open set A (0, T ): χ > 0 a.e. in A) µ = χ ε(u t), η = χ ε(u), E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

60 The degenerate problem (δ = 0): the existence theorem [E.R., R. Rossi, preprint arxiv: v1 (2012)] [Theorem 2] (δ = 0) Under the previous assumptions, there exist u W 1, (0, T ; L 2 ()) H 2 (0, T ; H 1 ()), µ L 2 (0, T ; L 2 ()), η L (0, T ; L 2 ()), w L r (0, T ; W 1,r ()) L (0, T ; L 1 ()) BV([0, T ]; W 1,r () ) χ L (0, T ; H s ()) H 1 (0, T ; L 2 ()), χ(x, t) 0, χ t(x, t) 0 a.e. such that it holds true (a.e. in any open set A (0, T ): χ > 0 a.e. in A) µ = χ ε(u t), η = χ ε(u), the weak enthalpy equation t t t ϕ(t) w(t)(dx) wϕ t dx + χ tθ(w)ϕ dx + K(w) w ϕ dx t = gϕ + w 0 ϕ(0) dx ϕ C 0 ([0, T ]; W 1,r ()) W 1,r (0, T ; L r ()), t (0, T ] 0 the weak momentum balance 2 t u div( χ µ) div( χ η)) = f in H 1 (; R d ), a.e. in (0, T ), and phase relations T T ( tχ + γ(χ)) ϕ dx T ( a s(χ, ϕ) 1 ) 0 2χ η 2 + Θ(w) ϕ dx for all ϕ L 2 (0, T ; H s +()) L (Q) with supp(ϕ) {χ > 0}. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

61 Degenerate total energy inequality For almost all t (0, T ] we obtain t H(t) + χ t 2 dx + 1 t µ 2 dx w 0 dx + 1 v 0 2 dx χ0 ε(u0) t t 2 as(χ0, χ0) + W (χ 0) dx + f u t dxdr + g dx 0 0 with H(t) w(t)(dx) + 1 tu(t) 2 dx as(χ(t), χ(t)) + W (χ(t)) dx + J (t) where J (t) := 1 lim inf η 2 δk (t) 2 dx δ k 0 t 1 with (η δk ) a suitable subsequence of (η δ ) from the approximated problem. And for all 0 t 1 t 2 T there holds t2 t2 ( H(r) dr w(r)(dx) + 1 ) 2 as(χ(r), χ(r)) dr + t 1 t2 t 1 ( ( 1 2 tu(r) 2 + W (χ(r)) + 1 ) ) 2 η(r) 2 dx dr. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

62 A comparison between the solution notions Weak solution to the degenerating system (δ = 0) when χ > 0 weak solution to the non-degenerating system (δ > 0) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

63 A comparison between the solution notions Weak solution to the degenerating system (δ = 0) when χ > 0 weak solution to the non-degenerating system (δ > 0) Suppose that the solution is more regular and χ > 0 a.e. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

64 A comparison between the solution notions Weak solution to the degenerating system (δ = 0) when χ > 0 weak solution to the non-degenerating system (δ > 0) Suppose that the solution is more regular and χ > 0 a.e. Then the following identities hold true: Hence µ = χ ε(u t), η = χ ε(u) a.e. in (0, T ) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

65 A comparison between the solution notions Weak solution to the degenerating system (δ = 0) when χ > 0 weak solution to the non-degenerating system (δ > 0) Suppose that the solution is more regular and χ > 0 a.e. Then the following identities hold true: Hence T 0 µ = χ ε(u t), η = χ ε(u) a.e. in (0, T ) T T ( ( tχ + γ(χ)) ϕ dx + a s(χ, ϕ) 1 ) 0 0 2χ η 2 ϕ + Θ(w)ϕ dx for all ϕ L 2 (0, T ; H s +()) L (Q) with supp(ϕ) {χ > 0} coincides with the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ ϕ L 2 (0, T ; H s + ()) L (Q) and with ξ I [0,+ ) (χ). E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

66 A comparison between the solution notions Weak solution to the degenerating system (δ = 0) when χ > 0 weak solution to the non-degenerating system (δ > 0) Suppose that the solution is more regular and χ > 0 a.e. Then the following identities hold true: Hence T 0 µ = χ ε(u t), η = χ ε(u) a.e. in (0, T ) T T ( ( tχ + γ(χ)) ϕ dx + a s(χ, ϕ) 1 ) 0 0 2χ η 2 ϕ + Θ(w)ϕ dx for all ϕ L 2 (0, T ; H s +()) L (Q) with supp(ϕ) {χ > 0} coincides with the one-sided inequality T χ tϕ + a s(χ, ϕ) + ξϕ + γ(χ)ϕ + ε(u) 2 ϕ Θ(w)ϕ ϕ L 2 (0, T ; H+ s ()) L (Q) and with ξ I [0,+ ) (χ). Subtracting from the degenerate total energy inequality the weak enthalpy equation tested by 1, we recover (a.e. in (0, T ]) the energy inequality for χ t χ t 2 dx dr as(χ(t), χ(t)) + W (χ(t)) dx 1 t ( ) 2 as(χ 0, χ 0 ) + W (χ 0 ) dx + χ t ε(u) 2 + Θ(w) dx dr 0 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

67 Work in progress: avoid the small perturbation assumptions We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

68 Work in progress: avoid the small perturbation assumptions We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 Our next aim: to couple the weak equations for u and χ with a suitable formulation of the internal energy balance: E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

69 Work in progress: avoid the small perturbation assumptions We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 Our next aim: to couple the weak equations for u and χ with a suitable formulation of the internal energy balance: a weak energy conservation and entropy inequality inspired by the works of E. Feireisl and co-authors ([Feireisl, Comput. Math. Appl. (2007)] and [Buĺıček, Feireisl, & Málek, Nonlinear Anal. Real World Appl. (2009)]) for heat conduction in fluids E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

70 Entropic formulation: a phase transitions model E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

71 A first application of the entropic formulation: solid-liquid phase transitions In order to show the potential power of this idea we... give a description of the method stating more precisely the content of this recent work [E. Feireisl, H. Petzeltovà, E.R., Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci. (2009)] in which this notion of solution has been firstly applied to phase transition models E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

72 A first application of the entropic formulation: solid-liquid phase transitions In order to show the potential power of this idea we... give a description of the method stating more precisely the content of this recent work [E. Feireisl, H. Petzeltovà, E.R., Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci. (2009)] in which this notion of solution has been firstly applied to phase transition models We consider there a model for solid-liquid phase transitions associated to a nonlinear PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

73 A first application of the entropic formulation: solid-liquid phase transitions In order to show the potential power of this idea we... give a description of the method stating more precisely the content of this recent work [E. Feireisl, H. Petzeltovà, E.R., Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci. (2009)] in which this notion of solution has been firstly applied to phase transition models We consider there a model for solid-liquid phase transitions associated to a nonlinear PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c No global-in-time well-posedness result had yet been obtained in the 3D case, even neglecting χ t 2 on the r.h.s. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

74 A first application of the entropic formulation: solid-liquid phase transitions In order to show the potential power of this idea we... give a description of the method stating more precisely the content of this recent work [E. Feireisl, H. Petzeltovà, E.R., Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci. (2009)] in which this notion of solution has been firstly applied to phase transition models We consider there a model for solid-liquid phase transitions associated to a nonlinear PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c No global-in-time well-posedness result had yet been obtained in the 3D case, even neglecting χ t 2 on the r.h.s. A 1D global result was proved in [F. Luterotti and U. Stefanelli, ZAA (2002)] E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

75 A first application of the entropic formulation: solid-liquid phase transitions In order to show the potential power of this idea we... give a description of the method stating more precisely the content of this recent work [E. Feireisl, H. Petzeltovà, E.R., Existence of solutions to some models of phase changes with microscopic movements, Math. Meth. Appl. Sci. (2009)] in which this notion of solution has been firstly applied to phase transition models We consider there a model for solid-liquid phase transitions associated to a nonlinear PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c No global-in-time well-posedness result had yet been obtained in the 3D case, even neglecting χ t 2 on the r.h.s. A 1D global result was proved in [F. Luterotti and U. Stefanelli, ZAA (2002)] = a new weaker notion of solution is needed E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

76 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

77 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the entropy of the system is controlled by dissipation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

78 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the entropy of the system is controlled by dissipation and E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

79 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the entropy of the system is controlled by dissipation and total energy is conserved during the evolution E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

80 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the entropy of the system is controlled by dissipation and total energy is conserved during the evolution The nonlinear equation for θ (internal energy balance) is replaced by the entropy inequality + the total energy conservation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

81 Entropic formulation Idea: Start directly from the basic principles of Thermodynamics just assuming that the entropy of the system is controlled by dissipation and total energy is conserved during the evolution The nonlinear equation for θ (internal energy balance) is replaced by the entropy inequality + the total energy conservation Finally, couple these relations to a suitable phase dynamics E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

82 The entropy production Assuming the system is thermally isolated, the entropy balance results E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

83 The entropy production Assuming the system is thermally isolated, the entropy balance results T T q T s tϕ θ ϕ = rϕ ϕ D(Q T ) 0 r represents the entropy production rate. 0 0 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

84 The entropy production Assuming the system is thermally isolated, the entropy balance results T T q T s tϕ θ ϕ = rϕ ϕ D(Q T ) 0 0 r represents the entropy production rate. Then, in order to comply with the Clausius-Duhem inequality, we assume: (i) r is a nonnegative measure on [0, T ] =: Q T ; (ii) r 1 ( χ t 2 q θ ) 0. θ θ 0 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

85 The entropy production Assuming the system is thermally isolated, the entropy balance results T T q T s tϕ θ ϕ = rϕ ϕ D(Q T ) 0 0 r represents the entropy production rate. Then, in order to comply with the Clausius-Duhem inequality, we assume: (i) r is a nonnegative measure on [0, T ] =: Q T ; (ii) r 1 ( χ t 2 q θ ) 0. θ θ Taking q = θ, s = log θ + χ, we get T ( ) (log θ + χ) tϕ log θ ϕ dxdt 0 for every test function ϕ D(Q T ), ϕ 0 0 T 1 ( ) χ t 2 log θ θ ϕdxdt 0 θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

86 The entropy production Assuming the system is thermally isolated, the entropy balance results T T q T s tϕ θ ϕ = rϕ ϕ D(Q T ) 0 0 r represents the entropy production rate. Then, in order to comply with the Clausius-Duhem inequality, we assume: (i) r is a nonnegative measure on [0, T ] =: Q T ; (ii) r 1 ( χ t 2 q θ ) 0. θ θ Taking q = θ, s = log θ + χ, we get T ( ) (log θ + χ) tϕ log θ ϕ dxdt 0 for every test function ϕ D(Q T ), ϕ 0 0 T 1 ( ) χ t 2 log θ θ ϕdxdt 0 θ the total entropy is controlled by dissipation E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

87 The energy conservation and phase relation The total energy has to be preserved. Hence E(t) = E(0) for a.e. t [0, T ] where E ) (θ + W (χ) + χ 2 dx. 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

88 The energy conservation and phase relation The total energy has to be preserved. Hence E(t) = E(0) for a.e. t [0, T ] where E ) (θ + W (χ) + χ 2 dx. 2 Finally, the phase dynamics results as χ t χ + W (χ) = θ θ c a.e. in (0, T ), where W is a double well or double obstacle potential: W = β + γ where β : R [0, + ] is proper, lower semi-continuous, convex function γ C 2 (R), γ C 0,1 (R) : γ (r) K for all r R, W (r) c w r 2 for all r dom( β) Examples: β(r) = r ln(r) + (1 r) ln(1 r) or β(r) = I [0,1] (r) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

89 The existence theorem [E. Feireisl, H. Petzeltová, E.R., M2AS (2009)] Fix T > 0 and take suitable initial data. Let s (1, 2) be a proper exponent depending on the space dimension. E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

90 The existence theorem [E. Feireisl, H. Petzeltová, E.R., M2AS (2009)] Fix T > 0 and take suitable initial data. Let s (1, 2) be a proper exponent depending on the space dimension.then there exists at least one pair (θ, χ) s.t. θ L (0, T ; L 1 ()) L s (Q T ), θ(x, t) > 0 a. e. in Q T log(θ) L (0, T ; L 1 ()) L 2 (0, T ; H 1 ()) W 1,1 (0, T ; W 2,3/2 ()) χ C 0 ([0, T ]; H 1 ()) L s (0, T ; W 2,s N ()) χt Ls (Q T ), E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

91 The existence theorem [E. Feireisl, H. Petzeltová, E.R., M2AS (2009)] Fix T > 0 and take suitable initial data. Let s (1, 2) be a proper exponent depending on the space dimension.then there exists at least one pair (θ, χ) s.t. θ L (0, T ; L 1 ()) L s (Q T ), θ(x, t) > 0 a. e. in Q T log(θ) L (0, T ; L 1 ()) L 2 (0, T ; H 1 ()) W 1,1 (0, T ; W 2,3/2 ()) χ C 0 ([0, T ]; H 1 ()) L s (0, T ; W 2,s N ()) χt Ls (Q T ), satisfying the entropy inequality ( ϕ D(Q T ), ϕ 0): T ((log θ + χ) tϕ log θ ϕ) dx dt 0 T 0 1 ( ) χ t 2 log θ θ ϕ dx dt θ E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

92 The existence theorem [E. Feireisl, H. Petzeltová, E.R., M2AS (2009)] Fix T > 0 and take suitable initial data. Let s (1, 2) be a proper exponent depending on the space dimension.then there exists at least one pair (θ, χ) s.t. θ L (0, T ; L 1 ()) L s (Q T ), θ(x, t) > 0 a. e. in Q T log(θ) L (0, T ; L 1 ()) L 2 (0, T ; H 1 ()) W 1,1 (0, T ; W 2,3/2 ()) χ C 0 ([0, T ]; H 1 ()) L s (0, T ; W 2,s N ()) χt Ls (Q T ), satisfying the entropy inequality ( ϕ D(Q T ), ϕ 0): T ((log θ + χ) tϕ log θ ϕ) dx dt the phase equation 0 T 0 1 ( ) χ t 2 log θ θ ϕ dx dt θ χ t χ + W (χ) = θ θ c a.e. in Q T, χ(0) = χ 0 a.e. in E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

93 The existence theorem [E. Feireisl, H. Petzeltová, E.R., M2AS (2009)] Fix T > 0 and take suitable initial data. Let s (1, 2) be a proper exponent depending on the space dimension.then there exists at least one pair (θ, χ) s.t. θ L (0, T ; L 1 ()) L s (Q T ), θ(x, t) > 0 a. e. in Q T log(θ) L (0, T ; L 1 ()) L 2 (0, T ; H 1 ()) W 1,1 (0, T ; W 2,3/2 ()) χ C 0 ([0, T ]; H 1 ()) L s (0, T ; W 2,s N ()) χt Ls (Q T ), satisfying the entropy inequality ( ϕ D(Q T ), ϕ 0): T ((log θ + χ) tϕ log θ ϕ) dx dt the phase equation 0 T 0 1 ( ) χ t 2 log θ θ ϕ dx dt θ χ t χ + W (χ) = θ θ c a.e. in Q T, χ(0) = χ 0 a.e. in and the total energy conservation E(t) = E(0) a.e. in [0, T ], E ) (θ + W (χ) + χ 2 dx 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

94 The main advantages of this approach E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

95 The main advantages of this approach It complies with thermodynamical principles and hence it gives for free thermodynamically consistent models E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

96 The main advantages of this approach It complies with thermodynamical principles and hence it gives for free thermodynamically consistent models It gives rise exactly to the previous the PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c at least in case the solution (θ, χ) is sufficiently smooth E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

97 The main advantages of this approach It complies with thermodynamical principles and hence it gives for free thermodynamically consistent models It gives rise exactly to the previous the PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c at least in case the solution (θ, χ) is sufficiently smooth However, in this case and similarly in many other situations, to prove that the solution has this extra regularity is out of reach E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

98 The main advantages of this approach It complies with thermodynamical principles and hence it gives for free thermodynamically consistent models It gives rise exactly to the previous the PDE system θ t + χ tθ θ = χ t 2 χ t χ + W (χ) = θ θ c at least in case the solution (θ, χ) is sufficiently smooth However, in this case and similarly in many other situations, to prove that the solution has this extra regularity is out of reach It can be suitable also in different applications such as the ones related to phase transitions in viscoelastic materials, SMA, liquid crystal flows, damage phenomena E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

99 Work in progress: an entropic formulation for the damage phenomena We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

100 Work in progress: an entropic formulation for the damage phenomena We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 Our next aim: to couple the weak equations for u and χ with the entropy production T ( ) (log θ + χ) tϕ log θ ϕ dxdt 0 T 1 ( ) χ t 2 χ ε(u t) 2 log θ θ ϕdxdt 0 θ for every test function ϕ D(Q T ), ϕ 0 E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

101 Work in progress: an entropic formulation for the damage phenomena We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 Our next aim: to couple the weak equations for u and χ with the entropy production T ( ) (log θ + χ) tϕ log θ ϕ dxdt 0 T 1 ( ) χ t 2 χ ε(u t) 2 log θ θ ϕdxdt 0 θ for every test function ϕ D(Q T ), ϕ 0 and the energy conservation E(t) = E(0) for a.e. t [0, T ], where ) 12 ut 2 E (θ + W (χ) + as(χ, χ) + + χ ε(u) 2 dx 2 2 This is a work in progress (with R. Rossi) E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

102 Work in progress: an entropic formulation for the damage phenomena We worked here with the small perturbation assumption, i.e. neglecting the quadratic contribution on the r.h.s in the internal energy balance: θ t + χ tθ θ = χ t 2 + χ ε(u t) 2 Our next aim: to couple the weak equations for u and χ with the entropy production T ( ) (log θ + χ) tϕ log θ ϕ dxdt 0 T 1 ( ) χ t 2 χ ε(u t) 2 log θ θ ϕdxdt 0 θ for every test function ϕ D(Q T ), ϕ 0 and the energy conservation E(t) = E(0) for a.e. t [0, T ], where ) 12 ut 2 E (θ + W (χ) + as(χ, χ) + + χ ε(u) 2 dx 2 2 This is a work in progress (with R. Rossi) Finally, with C. Heinemann, C. Kraus and R. Rossi, we would like to study the case of non-isothermal phase separation and damage E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32

103 A further application to liquid crystals In [E. Feireisl, M. Frémond, E.R., G. Schimperna, ARMA 2012] we have coupled the incompressible Nevier-Stokes equation div v = 0, S = ν(θ) ( xv + t xv ), v t + v xv + xp = div S + div σ nd + g σ nd = xd xd + ( d W (d) d) d where xd xd is the 3 3 matrix given by i d j d, (a b) ij := a i b j, 1 i, j 3, and the evolution of the director field d, representing preferred orientation of molecules in a neighborhood of any point of a reference domain d t + v xd d xv = d d W (d) with an entropic formulation of the inernal energy balance displaying higher order nonlinearities on the right hand side θ t + v θ + div q = S : xv+ d d W (d) 2 In [E. Feireisl, E.R., G. Schimperna, A. Zarnescu, preprint arxiv: v1 2012] we have extended it to the tensorial Ball-Majumdar model for liquid crystals E. Rocca (Università di Milano) Weak solutions for degenerating PDEs January 30, / 32