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1078-0947
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1553-5231
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Discrete & Continuous Dynamical Systems
June 2015 , Volume 35 , Issue 6
Special issue dedicated to Jürgen Sprekels on the occasion of his 65th birthday
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2015, 35(6): i-ii
doi: 10.3934/dcds.2015.35.6i
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Abstract:
This special volume is dedicated to Jürgen Sprekels on the occasion of his 65th birthday, in tribute to his important achievements in several theoretical and applied problems, especially in the fields of Partial Differential Equations, Optimal Control, Hysteresis, Thermodynamics and Phase Transitions, Mechanics of Solids.
For more information please click the “Full Text” above.
This special volume is dedicated to Jürgen Sprekels on the occasion of his 65th birthday, in tribute to his important achievements in several theoretical and applied problems, especially in the fields of Partial Differential Equations, Optimal Control, Hysteresis, Thermodynamics and Phase Transitions, Mechanics of Solids.
For more information please click the “Full Text” above.
2015, 35(6): 2325-2347
doi: 10.3934/dcds.2015.35.2325
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We present a finite element method for simulating complex free surface flow. The mathematical model and the numerical method take into account two-phase non-isothermal flow of an incompressible liquid and a gas phase, capillary forces at the interface of both fluids, Marangoni effects due to temperature variation of the interface and mass transport across the interface by evaporation/condensation. The method is applied to two examples from microgravity research, for which experimental data are available.
We present a finite element method for simulating complex free surface flow. The mathematical model and the numerical method take into account two-phase non-isothermal flow of an incompressible liquid and a gas phase, capillary forces at the interface of both fluids, Marangoni effects due to temperature variation of the interface and mass transport across the interface by evaporation/condensation. The method is applied to two examples from microgravity research, for which experimental data are available.
2015, 35(6): 2349-2403
doi: 10.3934/dcds.2015.35.2349
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We focus on a highly nonlinear evolutionary PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction.
We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-tim solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation of) the problem.
Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.
We focus on a highly nonlinear evolutionary PDE system describing volume processes coupled with surfaces processes in thermoviscoelasticity, featuring the quasi-static momentum balance, the equation for the unidirectional evolution of an internal variable on the surface, and the equations for the temperature in the bulk domain and the temperature on the surface. A significant example of our system occurs in the modeling for the unidirectional evolution of adhesion between a body and a rigid support, subject to thermal fluctuations and in contact with friction.
We investigate the related initial-boundary value problem, and in particular the issue of existence of global-in-tim solutions, on an abstract level. This allows us to highlight the analytical features of the problem and, at the same time, to exploit the tight coupling between the various equations in order to deduce suitable estimates on (an approximation of) the problem.
Our existence result is proved by passing to the limit in a carefully tailored approximate problem, and by extending the obtained local-in-time solution by means of a refined prolongation argument.
2015, 35(6): 2405-2421
doi: 10.3934/dcds.2015.35.2405
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Rate independent evolutions can be formulated as operators, called hysteresis operators, between suitable function spaces. In this paper, we present some results concerning the existence and the form of directional derivatives and of Hadamard derivatives of such operators in the scalar case, that is, when the driving (input) function is a scalar function.
Rate independent evolutions can be formulated as operators, called hysteresis operators, between suitable function spaces. In this paper, we present some results concerning the existence and the form of directional derivatives and of Hadamard derivatives of such operators in the scalar case, that is, when the driving (input) function is a scalar function.
2015, 35(6): 2423-2442
doi: 10.3934/dcds.2015.35.2423
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The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.
The paper deals with a phase field system of Cahn-Hilliard type. For positive viscosity coefficients, the authors prove an existence and uniqueness result and study the long time behavior of the solution by assuming the nonlinearities to be rather general. In a more restricted setting, the limit as the viscosity coefficients tend to zero is investigated as well.
2015, 35(6): 2443-2463
doi: 10.3934/dcds.2015.35.2443
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We consider the problem to optimise the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimisation principle of the free energy while the temperature is solving the heat equation with radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallisation, we can expect that the interface has a global representation as a graph. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second derivatives of the surface and for the surface temperature gradient.
We consider the problem to optimise the stationary temperature distribution and the equilibrium shape of the solid-liquid interface in a two-phase system subject to a temperature gradient. The interface satisfies the minimisation principle of the free energy while the temperature is solving the heat equation with radiation boundary conditions at the outer wall. Under the condition that the temperature gradient is uniformly negative in the direction of crystallisation, we can expect that the interface has a global representation as a graph. We reformulate this condition as a pointwise constraint on the gradient of the state, and we derive the first order optimality system for a class of objective functionals that account for the second derivatives of the surface and for the surface temperature gradient.
2015, 35(6): 2465-2495
doi: 10.3934/dcds.2015.35.2465
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We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.
We pursue the study of fatigue accumulation in an oscillating elastoplastic beam under the additional hypothesis that the material can partially recover by the effect of melting. The full system consists of the momentum and energy balance equations, an evolution equation for the fatigue rate, and a differential inclusion for the phase dynamics. The main result consists in proving the existence and uniqueness of a strong solution.
2015, 35(6): 2497-2522
doi: 10.3934/dcds.2015.35.2497
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We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity $u$ is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter $\psi$ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a priori estimates and showing weak sequential stability of families of approximating solutions.
We introduce a diffuse interface model describing the evolution of a mixture of two different viscous incompressible fluids of equal density. The main novelty of the present contribution consists in the fact that the effects of temperature on the flow are taken into account. In the mathematical model, the evolution of the velocity $u$ is ruled by the Navier-Stokes system with temperature-dependent viscosity, while the order parameter $\psi$ representing the concentration of one of the components of the fluid is assumed to satisfy a convective Cahn-Hilliard equation. The effects of the temperature are prescribed by a suitable form of the heat equation. However, due to quadratic forcing terms, this equation is replaced, in the weak formulation, by an equality representing energy conservation complemented with a differential inequality describing production of entropy. The main advantage of introducing this notion of solution is that, while the thermodynamical consistency is preserved, at the same time the energy-entropy formulation is more tractable mathematically. Indeed, global-in-time existence for the initial-boundary value problem associated to the weak formulation of the model is proved by deriving suitable a priori estimates and showing weak sequential stability of families of approximating solutions.
2015, 35(6): 2523-2538
doi: 10.3934/dcds.2015.35.2523
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This paper is concerned with a heat convection problem. We discuss it in the framework of parabolic variational inequalities. The problem is a system of a heat equation with convection and a Navier-Stokes variational inequality with temperature-dependent velocity constraint. Our problem is a sort of parabolic quasi-variational inequalities in the sense that the constraint set for the velocity field depends on the unknown temperature. We shall give an existence result of the heat convection problem in a weak sense, and show that under some additional constraint for temperature there exists a strong solution of the problem.
This paper is concerned with a heat convection problem. We discuss it in the framework of parabolic variational inequalities. The problem is a system of a heat equation with convection and a Navier-Stokes variational inequality with temperature-dependent velocity constraint. Our problem is a sort of parabolic quasi-variational inequalities in the sense that the constraint set for the velocity field depends on the unknown temperature. We shall give an existence result of the heat convection problem in a weak sense, and show that under some additional constraint for temperature there exists a strong solution of the problem.
2015, 35(6): 2539-2564
doi: 10.3934/dcds.2015.35.2539
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We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are Hölder continuous with respect to $\beta$.
We consider the modified phase-field crystal (MPFC) equation that has recently been proposed by P. Stefanovic et al. This is a variant of the phase-field crystal (PFC) equation, introduced by K.-R. Elder et al., which is characterized by the presence of an inertial term $\beta\phi_{tt}$. Here $\phi$ is the phase function standing for the number density of atoms and $\beta\geq 0$ is a relaxation time. The associated dynamical system for the MPFC equation with respect to the parameter $\beta$ is analyzed. More precisely, we establish the existence of a family of exponential attractors $\mathcal{M}_\beta$ that are Hölder continuous with respect to $\beta$.
2015, 35(6): 2565-2590
doi: 10.3934/dcds.2015.35.2565
+[Abstract](2617)
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In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [12]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
In this paper, we consider a coupled PDE system describing phase separation and damage phenomena in elastically stressed alloys in the presence of inertial effects. The material is considered on a bounded Lipschitz domain with mixed boundary conditions for the displacement variable. The main aim of this work is to establish existence of weak solutions for the introduced hyperbolic-parabolic system. To this end, we first adopt the notion of weak solutions introduced in [12]. Then we prove existence of weak solutions by means of regularization, time-discretization and different variational techniques.
2015, 35(6): 2591-2614
doi: 10.3934/dcds.2015.35.2591
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In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.
In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.
In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be continuous and piecewise monotaffine, i.e. being piecewise the composition of two functions such that the output of a monotone increasing function is used as input for an affine function.
In the current paper, a representation result is formulated for hysteresis operators dealing with input functions being left-continuous and piecewise monotaffine and continuous. The operators are generated by functions acting on an admissible subset of the set of all strings of pairs of elements of the vector space.
2015, 35(6): 2615-2623
doi: 10.3934/dcds.2015.35.2615
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We investigate a variational theory for magnetoelastic solids under the incompressibility constraint. The state of the system is described by deformation and magnetization. While the former is classically related to the reference configuration, magnetization is defined in the deformed configuration instead. We discuss the existence of energy minimizers without relying on higher-order deformation gradient terms. Then, by introducing a suitable positively $1$-homogeneous dissipation, a quasistatic evolution model is proposed and analyzed within the frame of energetic solvability.
We investigate a variational theory for magnetoelastic solids under the incompressibility constraint. The state of the system is described by deformation and magnetization. While the former is classically related to the reference configuration, magnetization is defined in the deformed configuration instead. We discuss the existence of energy minimizers without relying on higher-order deformation gradient terms. Then, by introducing a suitable positively $1$-homogeneous dissipation, a quasistatic evolution model is proposed and analyzed within the frame of energetic solvability.
2015, 35(6): 2625-2657
doi: 10.3934/dcds.2015.35.2625
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An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional.
The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated.
The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities.
The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.
An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional.
The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated.
The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities.
The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.
2015, 35(6): 2659-2677
doi: 10.3934/dcds.2015.35.2659
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We consider evolutionary reaction-diffusion problems with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and the exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.
We consider evolutionary reaction-diffusion problems with mixed Dirichlet--Robin boundary conditions. For this class of problems, we derive two-sided estimates of the distance between any function in the admissible energy space and the exact solution of the problem. The estimates (majorants and minorants) are explicitly computable and do not contain unknown functions or constants. Moreover, it is proved that the estimates are equivalent to the energy norm of the deviation from the exact solution.
2015, 35(6): 2679-2700
doi: 10.3934/dcds.2015.35.2679
+[Abstract](2861)
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We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary $\Gamma$-convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show $\Gamma$-convergence of the associated energies in suitable function spaces.
The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in $L^2$, while for the case of a quadratic nonlinearity we need to impose weak convergence in $H^1$. However, we do not need well-preparedness of the initial conditions.
We discuss the justification of the Ginzburg-Landau equation with real coefficients as an amplitude equation for the weakly unstable one-dimensional Swift-Hohenberg equation. In contrast to classical justification approaches we employ the method of evolutionary $\Gamma$-convergence by reformulating both equations as gradient systems. Using a suitable linear transformation we show $\Gamma$-convergence of the associated energies in suitable function spaces.
The limit passage of the time-dependent problem relies on the recent theory of evolutionary variational inequalities for families of uniformly convex functionals as developed by Daneri and Savaré 2010. In the case of a cubic energy it suffices that the initial conditions converge strongly in $L^2$, while for the case of a quadratic nonlinearity we need to impose weak convergence in $H^1$. However, we do not need well-preparedness of the initial conditions.
2015, 35(6): 2701-2710
doi: 10.3934/dcds.2015.35.2701
+[Abstract](2476)
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We introduce a general local parametrization for the solution of the implicit equation $f(x,y,z)=0$ by using Hamiltonian systems. The approach extends previous work of the authors and is valid in the critical case as well.
We introduce a general local parametrization for the solution of the implicit equation $f(x,y,z)=0$ by using Hamiltonian systems. The approach extends previous work of the authors and is valid in the critical case as well.
2015, 35(6): 2711-2739
doi: 10.3934/dcds.2015.35.2711
+[Abstract](2702)
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The goal of this paper is twofold. Firstly, we overview the known Flory--Huggins--de Gennes (FHdG) free energy and the associated degenerate singular Cahn--Hilliard--de Gennes (CHdG) model for isothermal phase separation in a binary polymer mixture. Secondly, motivated by the structure of the FHdG free energy, in which the gradient term is made up of energetic and entropic contributions, we set up a corresponding thermodynamically consistent model for nonisothermal phase separation in such mixture. The model is characterized by the modified both energy and entropy fluxes by suitable ``extra" terms. In this sense it generalizes the well-known Penrose--Fife model in which only entropy flux is modified by an ``extra" term.
The goal of this paper is twofold. Firstly, we overview the known Flory--Huggins--de Gennes (FHdG) free energy and the associated degenerate singular Cahn--Hilliard--de Gennes (CHdG) model for isothermal phase separation in a binary polymer mixture. Secondly, motivated by the structure of the FHdG free energy, in which the gradient term is made up of energetic and entropic contributions, we set up a corresponding thermodynamically consistent model for nonisothermal phase separation in such mixture. The model is characterized by the modified both energy and entropy fluxes by suitable ``extra" terms. In this sense it generalizes the well-known Penrose--Fife model in which only entropy flux is modified by an ``extra" term.
2015, 35(6): 2741-2761
doi: 10.3934/dcds.2015.35.2741
+[Abstract](3132)
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The aim of this paper is to prove an isoperimetric inequality relative to a convex domain $\Omega\subset\mathbb{R}^d$ intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius $r>0$ and the position $y\in\overline{\Omega}$ of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension $d,$ the diameter of the domain and the integrability exponent $p\in[1,d)$.
The aim of this paper is to prove an isoperimetric inequality relative to a convex domain $\Omega\subset\mathbb{R}^d$ intersected with balls with a uniform relative isoperimetric constant, independent of the size of the radius $r>0$ and the position $y\in\overline{\Omega}$ of the center of the ball. For this, uniform Sobolev, Poincaré and Poincaré-Sobolev inequalities are deduced for classes of (not necessarily convex) domains that satisfy a uniform cone property. It is shown that the constants in all of these inequalities solely depend on the dimensions of the cone, space dimension $d,$ the diameter of the domain and the integrability exponent $p\in[1,d)$.
2015, 35(6): 2763-2796
doi: 10.3934/dcds.2015.35.2763
+[Abstract](2561)
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The inclusion $\beta(u)\ni h$ in $V'$ is studied, assuming that $V$ is a reflexive Banach space, and that $\beta: V \to {\cal P}(V')$ is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242]. A notion of strict generalized pseudo-monotonicity is also introduced. The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional $V \!\times V'\to \mathbf{R} \cup \{+\infty\}$.
A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi's notion of $\Gamma$-convergence. The compactness and the convergence of the family of operators $\beta$ provide the (weak) structural stability of the inclusion $\beta(u)\ni h$ with respect to variations of $\beta$ and $h$, under the only assumptions that the $\beta$s are equi-coercive and the $h$s are equi-bounded.
These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form $D_t\partial\varphi(u) + \alpha(u) \ni h$, $\partial\varphi$ being the subdifferential of a convex lower semicontinuous mapping $\varphi$, and $\alpha$ a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.
The inclusion $\beta(u)\ni h$ in $V'$ is studied, assuming that $V$ is a reflexive Banach space, and that $\beta: V \to {\cal P}(V')$ is a generalized pseudo-monotone operator in the sense of Browder-Hess [MR 0365242]. A notion of strict generalized pseudo-monotonicity is also introduced. The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional $V \!\times V'\to \mathbf{R} \cup \{+\infty\}$.
A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi's notion of $\Gamma$-convergence. The compactness and the convergence of the family of operators $\beta$ provide the (weak) structural stability of the inclusion $\beta(u)\ni h$ with respect to variations of $\beta$ and $h$, under the only assumptions that the $\beta$s are equi-coercive and the $h$s are equi-bounded.
These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form $D_t\partial\varphi(u) + \alpha(u) \ni h$, $\partial\varphi$ being the subdifferential of a convex lower semicontinuous mapping $\varphi$, and $\alpha$ a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.
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