Handbook of Differential Equations: Evolutionary Equations (eBook)
608 Seiten
Elsevier Science (Verlag)
978-0-08-093197-5 (ISBN)
- Review of new results in the area
- Continuation of previous volumes in the handbook series covering Evolutionary PDEs
- Written by leading experts
The material collected in this volume discusses the present as well as expected future directions of development of the field with particular emphasis on applications. The seven survey articles present different topics in Evolutionary PDE's, written by leading experts.- Review of new results in the area- Continuation of previous volumes in the handbook series covering Evolutionary PDEs- Written by leading experts
Front cover 1
Handbook of Differential Equations: Evolutionary Equations 4
Copyright page 5
Preface 6
List of Contributors 8
Contents 10
Contents of Volume I 12
Contents of Volume II 14
Contents of Volume III 16
Chapter 1. Incompressible Euler Equations: The Blow-up Problem and Related Results 18
1. Introduction 20
2. Local well-posedness and blow-up criteria 27
3. Blow-up scenarios 37
4. Model problems 43
5. Dichotomy: singularity or global regular dynamics? 56
6. Spectral dynamics approach 60
7. Conservation laws for singular solutions 64
References 67
Chapter 2. Mathematical Methods in the Theory of Viscous Fluids 74
1. Balance laws 76
2. Formulation of basic physical principles 77
3. Constitutive theory 82
4. A priori estimates 86
5. Weak sequential stability 92
6. Long-time behavior 103
7. Singular limits 107
References 114
Chapter 3. Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains 120
1. Introduction 122
2. The global attractor 129
3. Exponential attractors 145
4. Nonautonomous systems 155
5. Dissipative PDEs in unbounded domains 166
6. Ill-posed dissipative systems and trajectory attractors 199
References 208
Chapter 4. The Cahn-Hilliard Equation 218
1. Introduction 220
2. Backwards diffusion and regularization 221
3. The Cahn-Hilliard equation and phase separation 223
4. Two prototype formulations 225
5. Existence, uniqueness, and regularity 228
6. Linear stability and spinodal decomposition 231
7. Comparison with experiment 233
8. Long time behavior and limiting motions 233
9. Upper bounds for coarsening 236
Acknowledgements 242
References 242
Chapter 5. Mathematical Analysis of Viscoelastic Fluids 246
1. The equations describing viscoelastic flows 248
2. Existence results for initial value problems 254
3. Development of singularities 256
4. Steady flows 259
5. Instabilities and change of type 267
6. Controllability of viscoelastic flows 272
7. Concluding remarks 273
References 274
Chapter 6. Application of Monotone Type Operators to Parabolic and Functional Parabolic PDE's 284
1. Introduction 286
2. Abstract Cauchy problem for first order evolution equations 287
3. Second order and higher order nonlinear parabolic differential equations 295
4. Parabolic functional differential equations containing functional dependence in lower order terms 304
5. Parabolic equations containing functional dependence in the main part 311
6. Parabolic functional differential equations in (0,) 319
7. Further applications 328
References 336
Chapter 7. Recent Results on Hydrodynamic Limits 340
1. Introduction 342
2. Fluid equations, relative entropy, and dissipative solutions 343
3. Kinetic equations 365
4. Hydrodynamic limits 373
5. Conclusion and open problems 389
Acknowledgements 390
References 390
Chapter 8. Introduction to Stefan-Type Problems 394
0. Introduction 398
1. The Stefan model 400
2. More general models of phase transitions 416
3. Analysis of the weak formulation of the Stefan model 438
4. Phase relaxation with nonlinear heat diffusion 457
5. Convexity and other analytical tools 466
Acknowledgments 483
6. Bibliography 484
References 485
Chapter 9. The KdV Equation 502
1. Historical background 504
2. The family of the KdV equations 508
3. The methods 510
4. Conservation laws 513
5. The KdV equation 514
6. The modified KdV equation 524
7. The potential KdV equation 534
8. The generalized KdV equation 540
9. The Gardner equation 543
10. Generalized KdV equation with two power nonlinearities 548
11. Fifth-order KdV equation 553
12. Seventh-order KdV equation 566
13. Ninth-order KdV equation 568
14. The coupled KdV or the Hirota-Satsuma equations 570
15. Compactons and the K(n,n) equation 576
16. Compacton-like solutions 582
References 582
Author Index 586
Subject Index 602
Chapter 2 Mathematical Methods in the Theory of Viscous Fluids
E. Feireisl
Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1, Czech Republic
1 Balance laws
Continuum mechanics describes a fluid in terms of observable and measurable macroscopic quantities: the density, the velocity, the absolute temperature, etc. The basic physical principles are expressed through balance laws that can be written in a general form:
(1.1)
where the symbol r denotes the volumic density, F is the flux, and s stands for the production rate of an observable quantity. Here, the time and the spatial position , where is the physical domain occupied by the fluid, play a role of independent reference variables, while B is an arbitrary subset of Ω. The reference system attached to the physical space corresponds to the Eulerian description of motion.
It is easy to check that (1.1) gives rise to
(1.2)
as soon as all quantities are continuously differentiable. However, the hypothesis of smoothness of the state variables is questionable, in particular in the case of the fluid density and other extensive quantities. Thus we should always keep in mind, that the “correct” formulation of a balance law is represented by the integral identity (1.1) rather than the partial differential equation (1.2).
On the other hand, given a vector field satisfying (1.2), we can define its normal trace on a space–time cylinder by means of the classical Gauss–Green theorem as
(1.3)
to be satisfied for all test functions .
Motivated by the previous discussion, we introduce a weak formulation of the balance law (1.1) as a family of integral identities
(1.4)
for any , where the production rate s can be a measure distributed on the set .
In accordance with formula (1.5), the measure s can capture the boundary behavior of the normal trace of the vector on the space–time cylinder , in particular, we recover the initial distribution , together with the boundary flux taking
(1.5)
where g is a (bounded) Radon measure on .
Thus relations (1.4), (1.5) can be formally interpreted as a partial differential equation
(1.6)
supplemented with the initial condition
(1.7)
and the boundary condition
(1.8)
Although the classical formulation (1.6)–(1.8) is widely used in the literature, the weak formulation expressed through (1.4), (1.5) seems to reflect better our understanding of macroscopic variables in continuum fluid mechanics as integral means rather than quantities that are well defined at each particular point of the underlying physical space. For further aspects of the weak formulation of conservation laws, the reader may consult the monograph by Dafermos [33], or a recent study by Chen and Torres [27].
2 Formulation of basic physical principles
Following the approach discussed in the previous section, we adopt the “weak” interpretation of the basic physical principles expressed through families of integral identities although they will be written in the classical way as a system of partial differential equations. Otherwise, the material presented below is classical and may be found in all standard texts devoted to continuum fluid mechanics: Batchelor [9], Chorin and Marsden [28], Gallavotti [61], Lamb [84], Lighthill [86], Truesdell [120,121,123], Truesdell and Rajagopal [122], among others.
2.1 Conservation of mass
The total mass of the fluid contained in a set at an instant t is given as
where stands for the density. Accordingly, the physical principle of mass conservation can be expressed in terms of the integral identity
(2.1)
where u denotes the velocity of the fluid. Equation (2.1) is supplemented with the initial condition
(2.2)
and the boundary conditions
(2.3)
As already pointed out, relations (2.1)–(2.3) are to be understood in the weak sense specified in (1.4)–(1.5).
2.2 Balance of momentum
Following the same line of arguments as in the preceding sections we can write the balance of momentum in the form
(2.4)
where denotes the Cauchy stress tensor, and f is a given external force. In addition, the fluids are characterized by Stokes' relation
(2.5)
where denotes the viscous stress tensor, and p is a scalar function termed pressure.
The initial distribution of momentum is given through
(2.6)
A proper choice of the boundary conditions for the fluid velocity offers more possibilities. Taking (2.3) for granted, we can assume that u satisfies the complete slip boundary condition
(2.7)
or, alternatively, the no-slip boundary condition
(2.8)
Note that both (2.7) and (2.8) are conservative in the sense that the kinetic energy flux vanishes on the boundary of Ω.
Similarly to (1.4), (1.5), the weak formulation of (2.4)–(2.6) reads
(2.9)
for any satisfying
If the no-slip boundary conditions (2.8) are imposed, we have to require, in addition, that
In contrast with the weak formulation of conservation laws introduced in the previous section, the satisfaction of the “vectorial” boundary conditions (2.3), (2.8) must be incorporated both in the choice of the functional space for u and the space of test functions. Furthermore, the no-slip boundary condition requires the existence of a trace of u on ∂Ω.
2.3 First law of thermodynamics, total energy balance
Formally, we can take the scalar product of (2.4) with u in order to deduce the kinetic energy balance
(2.10)
The first law of thermodynamics asserts that the energy of the fluid is a conserved quantity provided and there is no energy flux through the boundary. Accordingly, introducing the specific internal energy e we get the energy balance equation in the form
(2.11)
where the symbol q denotes the internal energy flux. If the system is energetically isolated, in particular if the boundary conditions (2.3), (2.7), or, alternatively (2.8), are supplemented with
(2.12)
Eq. (2.11) integrated over Ω gives rise to the total energy balance
(2.13)
2.4 Second law of thermodynamics, entropy
Subtracting (2.10) from (2.11) we obtain
(2.14)
which is an equation governing the time evolution of the internal energy.
In the absence of any dissipative mechanism in the system, meaning when , , Eq. (2.14) takes the form
(2.15)
The basic idea leading to the concept of entropy asserts that (2.15) can be written as a conservation law for a new state variable s termed entropy:
(2.16)
Now, assume that both e and s depend on and another internal variable ϑ called absolute temperature. Consequently, by help of (2.1), Eq. (2.15) can be written as
whence, necessarily,
where the factor H can be adjusted by a suitable choice of the temperature scale. Adopting the standard relation we arrive at Gibbs' equation
(2.17)
Accordingly, the internal energy balance (2.14) divided on ϑ gives...
| Erscheint lt. Verlag | 6.10.2008 |
|---|---|
| Sprache | englisch |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Analysis | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Technik | |
| ISBN-10 | 0-08-093197-9 / 0080931979 |
| ISBN-13 | 978-0-08-093197-5 / 9780080931975 |
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