Partial differential equation

A partial differential equation (abbreviation PDG , PDGL or PDGln , or PDE for English partial differential equation ) is a differential equation that contains partial derivatives . Such equations are used for the mathematical modeling of many physical processes. The solution theory of partial differential equations has been extensively researched for linear equations, but there are still many gaps for nonlinear equations. Numerical methods are usually used for the practical calculation of solutions .

introduction

Representation of a two-dimensional spherical wave

Some physical processes can be described by looking at the change in a variable with respect to a single variable. For example, the movement of a mass point in space is described by the equation of motion , which only contains derivatives with respect to time (namely speed and acceleration ). Such equations are called ordinary differential equations .

Many other physical processes can only be described if one considers the change in a variable with regard to several independent variables. If, for example at regular intervals a water drop on a water surface, the result is a spherical wave , as in the accompanying picture. The resulting wave depends on both the time derivative (speed of the wave) and the spatial derivative (profile of the wave). Since derivatives appear according to several variables, a partial differential equation is necessary to describe the process, the wave equation .

Representation of the solution of the one-dimensional transport equation for given initial values

A very simple partial differential equation is the linear transport equation in a space dimension. She has the shape

${\displaystyle {\frac {\partial u(x,t)}{\partial t}}+c{\frac {\partial u(x,t)}{\partial x}}=0}$

with a constant real parameter . ${\displaystyle c}$

The function you are looking for depends on two variables, where the location and time are usually designated. ${\displaystyle u(x,t)}$ ${\displaystyle x}$ ${\displaystyle t}$

Let us assume that the function is known at a certain time (approximately at that time ). So for everyone in the domain of definition, a relationship of the form is valid , whereby an arbitrarily given, at least once differentiable function is assumed ( initial condition ). ${\displaystyle u}$ ${\displaystyle t=0}$ ${\displaystyle x}$ ${\displaystyle u}$ ${\displaystyle u(x,0)=g(x)}$ ${\displaystyle g}$

Then the solution of the linear transport equation for arbitrary times is given by . This equation means nothing else than that the initial data is shifted ("transported") in unchanged form with the speed in the direction of the positive axis (along the characteristic of the equation), see adjacent picture. An application example would be the transport of a substance dissolved in the water with the flow of the water, for example the transport of pollutants in a river (whereby the diffusion of the substance is neglected). ${\displaystyle t}$ ${\displaystyle u(x,t)=g(x-ct)}$ ${\displaystyle g}$ ${\displaystyle c}$ ${\displaystyle x}$

Other examples of partial differential equations are

• the Poisson equation or the Laplace equation
• the heat conduction equation
• the Burgers equation .

For which applications are of great importance

• the Euler and Navier-Stokes equations
• the Maxwell equations
• the equations of magnetohydrodynamics
• the Schrödinger equation and
• the Korteweg-de-Vries equation .

definition

A partial differential equation is an equation (or system of equations) for one or more unknown functions that meets the following criteria:

• the unknown function depends on at least two variables (if it only depends on one variable, it is called an ordinary differential equation, or just differential equation for short)
• Partial derivatives occur in the equation for at least 2 variables
• only the function and its partial derivatives, each evaluated at the same point, appear in the equation.

The implicit form of a partial differential equation for a function that depends on two variables and is: ${\displaystyle u}$ ${\displaystyle x}$ ${\displaystyle y}$

${\displaystyle F\left(x,y,u(x,y),{\frac {\partial u(x,y)}{\partial x}},{\frac {\partial u(x,y)}{\partial y}},\ldots ,{\frac {\partial ^{2}u(x,y)}{\partial x\partial y}},\ldots \right)=0,}$

where is any function. In the multi-dimensional case ( describes a vector here ) one also writes ${\displaystyle F}$ ${\displaystyle x=(x_{1},\dotsc ,x_{n})^{t}}$

${\displaystyle F\left(x,u(x),\operatorname {D} u,\operatorname {D} ^{2}u,\ldots ,\operatorname {D} ^{k}u,\ldots \right)=0}$

with the partial derivatives of the degree . ${\displaystyle D^{k}u}$ ${\displaystyle k}$

Equations in which integrals occur in addition to partial derivatives are called integro-differential equations .

Classification

Partial differential equations can be classified according to various criteria. The solution behavior and thus the theoretical and numerical treatment of the equations classified in this way differ considerably from one another depending on the criterion used.

Number of leads

The degree of the highest derivative that occurs in the equation is called the order . For example, only partial first derivatives occur in a first order equation. In general, higher-order equations are more difficult to solve than lower-order equations.

Number of variables

In many partial differential equations, the number of variables plays a role in the possibilities of theoretical investigation and numerical solution. In the case of the Navier-Stokes equations , for example, extensive statements about existence, uniqueness and regularity could be proven in two spatial dimensions, while the question of the existence and uniqueness of smooth solutions in the three-dimensional case is open. This problem has been added to the list of Millennium Problems .

Linear and non-linear equations

One speaks of a linear partial differential equation if the unknown function and all derivatives occur linearly. This means that the coefficient functions in front of the unknown function or its derivatives depend only on the variables (and not on the function itself or its derivatives). So a linear partial differential equation of the 2nd order in two variables has the following general form: ${\displaystyle u}$

${\displaystyle {a(x,y){\frac {\partial ^{2}u(x,y)}{\partial x^{2}}}+b(x,y){\frac {\partial ^{2}u(x,y)}{\partial x\partial y}}+c(x,y){\frac {\partial ^{2}u(x,y)}{\partial y^{2}}}+d(x,y){\frac {\partial u(x,y)}{\partial x}}+e(x,y){\frac {\partial u(x,y)}{\partial y}}+f(x,y)u(x,y)=g(x,y)}}$.

The non-linear partial differential equation is called a quasilinear equation if all derivatives of the highest order occur linearly. An example is the minimum area equation :

${\displaystyle \left(1+{\frac {\partial }{\partial y}}u(x,y)^{2}\right){\frac {\partial ^{2}}{\partial x^{2}}}u(x,y)-2{\frac {\partial }{\partial x}}u(x,y){\frac {\partial }{\partial y}}u(x,y){\frac {\partial ^{2}}{\partial x\partial y}}u(x,y)+\left(1+{\frac {\partial }{\partial x}}u(x,y)^{2}\right){\frac {\partial ^{2}}{\partial y^{2}}}u(x,y)=0}$

or alternatively written:

${\displaystyle \left(1+u_{y}^{2}\right)u_{xx}-2u_{x}u_{y}u_{xy}+\left(1+u_{x}^{2}\right)u_{yy}=0}$

One speaks of a semilinear partial differential equation if in a quasilinear partial differential equation the coefficient functions before the highest derivative do not depend on lower derivatives and the unknown function. An example is the Korteweg-de-Vries equation :

${\displaystyle u_{yyy}+u_{x}+6u\cdot u_{y}=0}$.

An example of a nonlinear partial differential equation that is not quasi-linear is the Monge-Ampère equation :

${\displaystyle u_{xx}u_{yy}-u_{xy}^{2}=f}$.

Nonlinear equations usually describe much more complex phenomena than linear equations, such as turbulent flows (compared to laminar flows ). Nonlinear problems are more difficult to deal with than linear problems from both a theoretical and numerical point of view. A simple example of a nonlinear partial differential equation is the Burgers equation :

${\displaystyle {\frac {\partial u(x,t)}{\partial t}}+u(x,t){\frac {\partial u(x,t)}{\partial x}}=0}$.

Because its solutions are fully known, it often serves as a model problem for more general nonlinear equations, such as the Euler equations .

Basic types

Often partial differential equations are assigned to one of the three basic types elliptical , parabolic or hyperbolic . This classification is not exhaustive; so not every equation can be assigned to one of these types. It makes sense, however, because many equations that occur in practice fall into this scheme and the three basic types have fundamentally different properties. The classification is first described for the important case of a single second order equation.

2nd order equations

Two variables

As an example of the division into the three basic types elliptical, parabolic and hyperbolic differential equations, we consider the general linear partial differential equation of the 2nd order with two variables

${\displaystyle {a(x,y){\frac {\partial ^{2}u(x,y)}{\partial x^{2}}}+b(x,y){\frac {\partial ^{2}u(x,y)}{\partial x\partial y}}+c(x,y){\frac {\partial ^{2}u(x,y)}{\partial y^{2}}}+d(x,y){\frac {\partial u(x,y)}{\partial x}}+e(x,y){\frac {\partial u(x,y)}{\partial y}}+f(x,y)\,u(x,y)=0}}$

and define . ${\displaystyle D(x,y):=a(x,y)\,c(x,y)-\left({\tfrac {b(x,y)}{2}}\right)^{2}}$

Applies now

• ${\displaystyle D(x,y)>0}$, so the equation is called elliptic at the point , ${\displaystyle (x,y)}$
• ${\displaystyle D(x,y)=0}$, so the equation is called parabolic in the point , ${\displaystyle (x,y)}$
• ${\displaystyle D(x,y)<0}$, so the equation is called hyperbolic in the point . ${\displaystyle (x,y)}$

In this classification, only the coefficient functions , , the highest derivatives considered. Since they depend on the location , the type of differential equation also depends on the location. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$ ${\displaystyle (x,y)}$

The above definition gives the ellipticity of Poisson's equation , the parabolicity of the heat conduction equation and the hyperbolicity of the wave equation . These three equations each represent the normal type of all equations of their basic type.

The division into elliptic, parabolic and hyperbolic defined above can also be obtained by considering the definiteness of the coefficient matrix :

${\displaystyle A(x,y):={\begin{pmatrix}a(x,y)&{\frac {b(x,y)}{2}}\\{\frac {b(x,y)}{2}}&c(x,y)\end{pmatrix}}}$.

Is

• ${\displaystyle A(x,y)}$positive definite or negative definite, the equation is called elliptic in the point , ${\displaystyle (x,y)}$
• ${\displaystyle A(x,y)}$positive semidefinite or negative semidefinite, but not definite (singular), the equation is called parabolic in the point , ${\displaystyle (x,y)}$
• ${\displaystyle A(x,y)}$indefinite (with exactly one negative eigenvalue ), the equation is called hyperbolic in the point . ${\displaystyle (x,y)}$

A linear equation of the second order in two unknowns with real, constant coefficients can be assigned to exactly one of these types. As soon as the coefficients are not constant or the equation is non-linear, there are also equations that cannot be classified according to this scheme. The same applies to the more general cases described below. ${\displaystyle (x,y)}$

The origin of the terms elliptical , parabolic and hyperbolic arises from the theory of conic sections . The general conic equation

${\displaystyle a\,x^{2}+b\,x\,y+c\,y^{2}+d\,x+e\,y+f=0}$

has a similar structure to the linear partial differential equation of the 2nd order given above. For the coefficients , , analogous conditions apply as above, so that one from the corresponding conic sections ellipse , parabola or hyperbole created. ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle c}$

n variables

The type classification based on the coefficient matrix can also be transferred to equations of order 2 with more than two variables. In this case, the differential equation is used to create a matrix with the coefficient functions of the partial derivatives of the second order as entries. In generalizing the above case, the following applies: ${\displaystyle A({\vec {x}})}$

Is

• ${\displaystyle A({\vec {x}})}$positive definite or negative definite, the equation is called elliptic in the point , ${\displaystyle {\vec {x}}}$
• ${\displaystyle A({\vec {x}})}$semidefinite with zero as a simple eigenvalue, the equation is called parabolic in the point , ${\displaystyle {\vec {x}}}$
• ${\displaystyle A({\vec {x}})}$indefinite with eigenvalues ​​different from zero, the equation is called hyperbolic in the point . ${\displaystyle {\vec {x}}}$

Edge and initial value problems

The solution of a partial differential equation is generally not uniquely determined. In order to obtain a clear solution, certain additional conditions are required, namely initial and / or boundary conditions . In contrast to the situation with ordinary differential equations , however, only a selection of the initial and boundary conditions that is adapted to the respective basic type leads to a correctly posed problem .

Typical examples of correctly posed problems are:

• for elliptical problems:
  • Dirichlet boundary conditions
  • Neumann boundary conditions
  • inclined boundary conditions (also: Robin boundary conditions)
• Initial and boundary conditions for parabolic problems
• Initial conditions for hyperbolic problems.

theory

Methods of functional analysis

While with ordinary differential equations the problem of existence and uniqueness of the solution is solved in a very satisfactory way by Picard-Lindelof's theorem , with partial differential equations there is no such extensive general solution theory. Although Cauchy-Kowalewskaja's theorem guarantees the local existence and uniqueness of the solution of partial differential equations with analytic coefficient functions, this result cannot be extended to more general coefficient functions. There is already a counterexample for non-analytical coefficient functions that can be differentiated as often as desired, Lewy's example .

Since there is no satisfactory uniform theory of partial differential equations, these are divided into different types depending on the solution behavior . These are analyzed using different techniques in order to obtain information about the existence, uniqueness and other properties of the solutions. Linear partial differential equations have also been sufficiently well investigated in the case of multidimensional systems; this does not apply to non-linear partial differential equations.

In the theoretical investigation of the solutions to partial differential equations, as long as one only searches for classical (i.e. sufficiently often differentiable) solutions, one very quickly encounters great difficulties in mathematical theory. In addition, in some cases (e.g. when describing shock waves ), for physical reasons, the existence of continuous or differentiable solutions cannot be expected. For these reasons, in the classical theory, in many cases no or no sufficiently good statements about existence and uniqueness are possible.

As a way out, the concept of "solving a differential equation" is appropriately weakened; H. one also allows solutions that (in the classical sense) cannot be differentiated. With these extended solution concepts, sufficiently good theoretical statements are now possible in the weak theory . Furthermore, this weakened solution concept forms the basis of many numerical methods for the approximate solution of partial differential equations .

When examining partial differential equations, different solution terms appear:

• Classic solution : The solution is often continuously differentiable enough , and the equation is fulfilled by inserting these derivatives in every point of the underlying area.
• strong solution : The solution is often differentiable in the sense of the weak derivative , and the equation is fulfilled almost everywhere by inserting the weak derivatives .
• Weak solution : Here the equation ismultipliedby test functions , integrated and then at least partially integrated . A function from a suitable function space (usually a Sobolev space ) that satisfies this new equation for all test functions is called a weak solution.
• mild solution : Strong solutions often satisfy some form of variation of the constant formula. A solution to this formula is called a mild solution. However, a mild solution doesn't have to be a strong solution.
• Viscosity solution: Solutions of some types of equations can be constructed by considering a perturbed equation that is easier to solve with a small, additional diffusive or viscous term of a higher order and considering the limiting case in which this perturbation (the viscosity ) approaches zero. This leads to the concept of viscosity solution.
• Entropy solution: For some equations, the uniqueness is lost when changing to weak solutions , but it can be restored by adding an additional entropy condition. Such solutions are called entropy solutions. The name is motivated by the role of entropy in gas dynamic equations.
• Dimensional solution : For certain classes of nonlinear equations, a dimensional theoretical solution concept is useful in order to be able to describe possible concentration effects .
• distributional solution : The solution is a distribution and fulfills the equation in terms of distribution theory. All derivations are"circulated" to the test functions assumed tobe arbitrarily smooth . Since non-linear operations are generally not defined on distributions, this concept of solution is only useful for linear equations.

These terms are not used uniformly in the literature, so that reference must always be made to the respective definition.

With the help of regularity theory and Sobolew's embedding theorems , one can often show that the distributional or weak solution obtained is also a strong or even classical solution under suitable conditions for the differential equation.

Lie theory

→ Main article : Lie theory

A structured general approach to solving differential equations is pursued through symmetry and continuous group theory .

In 1870, Sophus Lie put the theory of differential equations on a generally applicable basis with the Lie theory. He showed that the older mathematical theories for solving differential equations can be summarized by the introduction of Lie groups .

A general approach to solving differential equations takes advantage of the symmetry property of differential equations. Continuous infinitesimal transformations are used that map solutions to (other) solutions of the differential equation. Symmetry methods are used to solve differential equations exactly.

Continuous group theory, Lie algebras and differential geometry are used to grasp the deeper structure of the linear and nonlinear (partial) differential equations and to map the relationships. See also the topics Lax Pairs , Recursive Operators, Kontakt and Bäcklund Transformations , which ultimately lead to the exact analytical solutions of a differential equation.

Basic types

For the classification, see section Classification according to basic type .

Elliptic partial differential equations

→ Main article : Elliptic partial differential equation

Solution of the Laplace equation on a circular ring with the Dirichlet boundary values ​​u (r = 2) = 0 and u (r = 4) = 4sin (5 * θ)

Elliptic partial differential equations typically occur in connection with time-independent ( stationary ) problems. A characteristic is that elliptic equations often describe a state of minimal energy , i.e. they result from the variation problem of a downwardly restricted effect.

The most famous examples are the Laplace equation and the Poisson equation . They describe, for example, the (stationary) temperature distribution or the electrostatic charge distribution , each in a body. In addition, the (Newtonian) gravitational potential is a solution to Poisson's equation. A non-linear elliptical equation is the equation for minimal surfaces (minimal surface equation), the shape of a soap film describes that forms upon immersion of a wire frame in soapy water.

For elliptic equations, the most common boundary conditions are:

• Dirichlet boundary conditions (the values ​​of the function sought are given on the boundary)
• Neumann boundary conditions (specification of the normal derivative of the function sought).

The difference is shown using the example of temperature distribution: If you put an object in ice water , the temperature at the edge is 0 degrees Celsius. The temperature distribution inside the object thus corresponds to the solution of a Dirichlet boundary value problem. Another case occurs when one isolates the body . The temperature is unknown here, but due to the insulation the heat flow at the edge is 0. Since the flow can be related to the normal derivation, this leads to a Neumann problem.
The same applies to electrostatics : if you know the voltage that is applied at the edge, you get a Dirichlet problem, if you know the current strength at the edge, you get a Neumann problem.

Parabolic partial differential equations

Solution of a two-dimensional heat equation

→ Main article : Parabolic partial differential equation

This type of equation describes phenomena similar to elliptical equations, but in the unsteady case.

By far the most important example of a parabolic equation is the heat conduction equation , which describes the cooling and heating of a body. Diffusion processes are also described by this equation. Another, non-linear example of a parabolic equation is the Korteweg-de-Vries equation , which describes water waves near the shore.

Parabolic equations require spatial boundary conditions and initial values ​​in time. For example, in the heat conduction equation, either the temperature or the temperature flow must be specified for all times at the (spatial) edge of the area; this corresponds to the Dirichlet or Neumann conditions in the elliptical case. In addition, the temperature distribution must be specified at the beginning, i.e. at the point in time . ${\displaystyle t=0}$

Hyperbolic partial differential equations

Vibrating membrane as a solution to a two-dimensional wave equation

The typical hyperbolic equation is the wave equation . In general, these types of equations describe waves and their propagation. In addition, individual first-order equations are always hyperbolic (in contrast to first-order systems, where different cases are again possible). In contrast to parabolic and elliptical equations, solutions of hyperbolic equations are dampened little or not at all . On the one hand, this leads to a complicated solution theory, since less differentiability can be expected. On the other hand, waves can only propagate over long distances through this lack of attenuation.

The initial and marginal values ​​belonging to this type lead to Cauchy problems . This means that, as in the parabolic case, initial values ​​are required in addition to spatial boundary conditions. To solve second-order hyperbolic equations, two initial values ​​are required - the function value and its time derivative at the beginning.

This should be illustrated using the example of a clamped string : The deflection of the string fulfills the wave equation. If the string is clamped at the ends, this leads to the spatial boundary conditions, in this case the deflection at the edge is 0 (because it is clamped). This means that the function value at the edge is known and Dirichlet boundary conditions result. (In the case of freely vibrating objects, such as the air column in woodwind instruments , one arrives at Neumann boundary conditions accordingly.) In addition, two initial conditions must now be specified: the deflection at the beginning (corresponds to the function value) and the speed with which the string at the beginning is plucked (corresponds to the time derivative). With these conditions, the deflection can be clearly specified at all later times.

Hyperbolic equations with pairwise different eigenvalues ​​are called strictly hyperbolic . Here the solution theory is also known for nonlinear systems . If the equations are not strictly hyperbolic, such as the multidimensional Euler equations or the equations of magnetohydrodynamics , this is no longer the case.

Numerical methods

The most widely used numerical methods are the finite element (FEM), finite differences (FDM) and finite volumes (FVM) methods . In recent years, the edge element method (REM or BEM) has also been used increasingly .

All these methods are based on a discretization of the partial differential equation, in which the solution is approximated with the help of a finite-dimensional structure.

literature

Historical literature

• Heinrich Weber : The partial differential equations of mathematical physics according to Riemann's lectures , Braunschweig: F. Vieweg and sohn, 1900–01.
• Jakob Horn : Introduction to the theory of partial differential equations , Leipzig: GJ Göschen, 1910.
• Richard Courant and David Hilbert : Methods of Mathematical Physics , Volume Two, Berlin: J. Springer, 1937.
• Olga Alexandrowna Ladyschenskaja , Nina Nikolajewna Uralzewa : Linear and Quasilinear Equations of Elliptic Type . Translated by L. Ehrenpreis, Academic Press, New York / London 1968, ISBN 0-12-432850-4 .
• Olga Alexandrowna Ladyschenskaja, WA Solonnikow, Nina N. Uralzewa: Linear and Quasi-linear Equations of Parabolic Type . Translated by S. Smith, American Mathematical Society, Providence, Rhode Island, 1968, ISBN 0-8218-1573-3 .

Modern literature

• David Gilbarg , Neil Trudinger : Elliptic Partial Differential Equations of Second Order. In: Basic Teachings of the Mathematical Sciences. Volume 224, Springer-Verlag, Berlin / Heidelberg / New York 1977, ISBN 3-540-08007-4 .
• Lawrence Craig Evans : Partial Differential Equations. In: Graduate Studies in Mathematics. Volume 19, American Mathematical Society, Providence, Rhode Island, 1999, ISBN 0-8218-0772-2 .
• Gerhard Dziuk: Theory and Numerics of Partial Differential Equations. de Gruyter, Berlin 2010, ISBN 978-3-11-014843-5 .
• Fritz John : Partial Differential Equations , 4th edition, Springer Verlag 1982, 1991, ISBN 0-387-90609-6
• Jürgen Jost : Partial differential equations. Springer, Berlin, 1998, ISBN 3-540-64222-6 .
• Randall J. LeVeque: Finite volume methods for hyperbolic problems. Cambridge Univ. Press, Cambridge 2002, ISBN 0-521-00924-3 .
• T. Roubíček: Nonlinear Partial Differential Equations with Applications. Birkhäuser, Basel, 2nd Ed .: 2013, ISBN 978-3-0348-0512-4 , doi: 10.1007 / 978-3-0348-0513-1 .
• B. Schweizer: Partial differential equations. Springer, Heidelberg 2013, ISBN 978-3-642-40637-9 .

Literature on Lie theory

• Nail H. Ibragimov CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3 CRC-Press 1993 ISBN 0-8493-4488-3 .
• Hans Stephani Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum, Cambridge University Press 1989.
• Peter Olver 1995 Equivalence, Invariants and Symmetry Cambridge Press 1995

Individual evidence

1. ↑ Lawrence C. Evans: Partial Differential Equations . Providence (RI), 1999, p. 18
2. ↑ Lawrence C. Evans: Partial Differential Equations . Providence (RI), 1999, p. 9.
3. ↑ Jürgen Jost: Partial differential equations . Elliptical (and parabolic) equations (=  masterclass ). Springer-Verlag, Berlin Heidelberg 1998, ISBN 978-3-642-58888-4 , p. 3 ff ., doi : 10.1007 / 978-3-642-58888-4 .
4. ^ Hans Lewy: An example of a smooth linear partial differential equation without solution . In: The Annals of Mathematics . Vol. 66, No. 1 (July 1957), pp. 155-158.
5. ↑ Lawrence C. Evans: Partial Differential Equations (=  Graduate Studies in Mathematics . Volume 19 ). 3. Edition. American Mathematical Society, 2002, ISBN 0-8218-0772-2 , ISSN  1065-7339 , Section 1.3 Strategies for Studying PDEs , p. 7-9 .
6. ↑ Ben Schweizer: Partial differential equations . An application-oriented introduction. 1st edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-40637-9 , solution terms in section 3.3 Sobolev spaces , p. 62-64 , doi : 10.1007 / 978-3-642-40638-6 .
7. ↑ Helmut Fischer, Helmut Kaul: Mathematics for Physicists, Volume 2 . Ordinary and partial differential equations, mathematical fundamentals of quantum mechanics. 4th edition. Springer Fachmedien, Wiesbaden 2014, ISBN 978-3-658-00476-7 , §13.1 Weak solutions of differential equations, p. 303-305 , doi : 10.1007 / 978-3-658-00477-4 .
8. ^ Dan Henry: Geometric theory of semilinear parabolic equations (=  Lecture notes in mathematics . Volume 840 ). 1st edition. Springer-Verlag, 1981, ISBN 978-3-540-10557-2 , ISSN  0075-8434 , Section 3.3: Local existence and uniqueness, p. 55 , doi : 10.1007 / BFb0089647 .
9. ↑ Klaus-Jochen Engel, Rainer Nagel et al .: One-Parameter Semigroups for Linear Evolution Equations (=  Graduate texts in mathematics . Volume 194 ). 1st edition. Springer-Verlag, 2000, ISBN 978-1-4757-7409-2 , ISSN  0072-5285 , 6. Well-Posedness for Evolution Equations, Definition 6.3, p. 146 , doi : 10.1007 / b97696 .
10. ↑ Lawrence C. Evans: Partial Differential Equations (=  Graduate Studies in Mathematics . Volume 19 ). 3. Edition. American Mathematical Society, 2002, ISBN 0-8218-0772-2 , ISSN  1065-7339 , Section 10.1 Introduction, Viscosity Solutions , p. 539-546 .
11. ↑ Lawrence C. Evans: Partial Differential Equations (=  Graduate Studies in Mathematics . Volume 19 ). 3. Edition. American Mathematical Society, 2002, ISBN 0-8218-0772-2 , ISSN  1065-7339 , Section 3.4 Introduction to Conservation Laws and Section 11.4 Entropy Criteria , p. 136-162, 599-611 .
12. ↑ Ronald J. DiPerna: Measure-valued solutions to conservation laws . In: C. Dafermos (Ed.): Archive for Rational Mechanics and Analysis . tape 88 , no. 3 . Springer-Verlag, September 1985, ISSN  1432-0673 , p. 223-270 , doi : 10.1007 / BF00752112 .
13. ↑ J. Málek, J. Nečas, M. Rokyta, M. Růžička: Weak and Measure-valued Solutions to Evolutionary PDEs (=  Applied mathematics and mathematical computation . Volume 13 ). 1st edition. Chapman and Hall, London Weinheim 1996, ISBN 978-0-412-57750-5 .