the heat equation - chennai mathematical institute

The Heat Equation

Vipul Naik

Basic properties ofthe heat equation

Physical intuitionbehind the heatequation

Properties of theheat equation

The generalconcept of flowand fixed points

Solving the heatequation in onevariable

Variations on theheat equation

Maximumprinciples

Why the heatequation matters

The Heat Equation

Vipul Naik

April 9, 2007

The Heat Equation

Vipul Naik


Partial derivatives

The heat equation inone dimension

Heat equation in morethan one dimension

The obviousproperties of the heatequation






Maximumprinciples


OutlineBasic properties of the heat equation

Partial derivativesThe heat equation in one dimensionHeat equation in more than one dimensionThe obvious properties of the heat equation

Physical intuition behind the heat equationHeat and its flowTemperature and two laws relating temperature to heat

Properties of the heat equationRecapitulationObvious structural propertiesWhat improves and what is unchanged

The general concept of flow and fixed pointsFlow and time-dependent vector fieldPicture of the flowThe heat equation as a flow

Solving the heat equation in one variableSeparation of variables

Variations on the heat equationThe heat equation as a diffusion equationGradient and reaction termsSupersolutions and subsolutions

Maximum principlesMaximum principle for the heat equationEffect of a reaction term

Why the heat equation mattersIn the physical worldIn the mathematical world

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


A quick recall of partial derivatives

Suppose u = u(x1, x2, . . . , xn) is a function Rn → R. Thenwe define:

∂u

∂xi:= (x1, x2, . . . , xn) 7→ (d/dxi )(xi 7→ u(x1, x2, . . . , xn))

In other words, the partial derivative in xi equals thederivative when viewed as a function of xi keeping the othervariables constant.Note that each ∂u

∂xiis also a map from Rn to R, viz it

evaluates at any point in Rn to give a real number.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Higher order partial derivativesIn addition to making sense of the first partials, we can alsomake sense of higher partials. To do this, observe that eachpartial is also a function from Rn to R, and can hence bedifferentiated in its own right.

Interestingly, we have the result:

∂ ∂u∂xi

∂xj=

∂ ∂u∂xj

∂xi

provided both sides are continuous functions. This is (aweak form of) Fubini’s theorem.We can thus simplify the notation and write:

∂2u

∂xi∂xj

In the particular case where i = j , we can write as:

∂2u

(∂xi )2

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Higher order partial derivativesIn addition to making sense of the first partials, we can alsomake sense of higher partials. To do this, observe that eachpartial is also a function from Rn to R, and can hence bedifferentiated in its own right.Interestingly, we have the result:

∂ ∂u∂xi

∂xj=

∂ ∂u∂xj

∂xi

provided both sides are continuous functions. This is (aweak form of) Fubini’s theorem.We can thus simplify the notation and write:

∂2u

∂xi∂xj

In the particular case where i = j , we can write as:

∂2u

(∂xi )2

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The actual equation

The heat equation is a differential equation involving threevariables – two independent variables x and t, and onedependent variable u = u(t, x).The equation states:

∂u

∂t= k

∂2u

(∂x)2

k ∈ R is a real number.

Here the symbol ∂u/∂t means the derivative of u withrespect to t, keeping x constant, while the symbol∂2u/(∂x)2 denotes the second partial derivative in x ,keeping t constant.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The actual equation

The heat equation is a differential equation involving threevariables – two independent variables x and t, and onedependent variable u = u(t, x).The equation states:

∂u

∂t= k

∂2u

(∂x)2

k ∈ R is a real number.Here the symbol ∂u/∂t means the derivative of u withrespect to t, keeping x constant, while the symbol∂2u/(∂x)2 denotes the second partial derivative in x ,keeping t constant.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Terminology associated with the heat equation

For the heat equation, we use the following terminology(which shall be clear once we get to the physical motivation):

1. The variable t is termed the time parameter, or thetime variable.

2. The variable x is termed the spatial parameter, or thespatial variable.

3. The function u is termed the heat function

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The n-dimensional heat equation

The heat equation in n dimensions is defined as follows:There are n + 1 independent variables, namely t (the timeparameter) and x1, x2, . . . , xn (the space parameters), andone dependent variable u = u(t, x1, x2, . . . , xn), subject to:

∂u

∂t= k

n∑i=1

∂2u

(∂xi )2

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The Laplacian operator

The Laplacian operator(defined) is a second-order differentialoperator that takes as input a function f : Rn → R, andoutputs another function, ∆f : Rn → R, where:

∆f =n∑

i=1

∂2f

(∂xi )2

Note that the Laplacian operator, as expressed in this form,appears to be heavily coordinate-dependent – a change ofbasis would change the Laplacian.

However, it turns out that any change of basis by anorthogonal matrix does not alter the value of the Laplacian.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The Laplacian operator

The Laplacian operator(defined) is a second-order differentialoperator that takes as input a function f : Rn → R, andoutputs another function, ∆f : Rn → R, where:

∆f =n∑

i=1

∂2f

(∂xi )2

Note that the Laplacian operator, as expressed in this form,appears to be heavily coordinate-dependent – a change ofbasis would change the Laplacian.However, it turns out that any change of basis by anorthogonal matrix does not alter the value of the Laplacian.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The heat equation using the Laplacian operator

We now think of u as a function R× Rn → R, where thefirst R is the time coordinate and the remaining Rn is thespace coordinate. We then have:

∂u

∂t= k∆u

where the Laplacian on the right side is taken only in termsof the space coordinates (for a fixed time).

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Autonomous nature

The heat equation is an autonomous differential equation. Inother words, the only way the dependent variables areinvoked is through differentiation – none of them appearexplicitly in the differential equation.

This means, in particular, that the heat equation is invariantunder both spatial translation and temporal translation.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Autonomous nature

The heat equation is an autonomous differential equation. Inother words, the only way the dependent variables areinvoked is through differentiation – none of them appearexplicitly in the differential equation.This means, in particular, that the heat equation is invariantunder both spatial translation and temporal translation.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


Invariance under orthogonal transformations

The heat equation states:

∂u

∂t= ∆u

Now, since ∆ (the Laplacian) is invariant under orthogonaltransformations, the overall heat equation is also invariantunder orthogonal transformations.

The Heat Equation

Vipul Naik


Partial derivatives









Maximumprinciples


The heat equation could thus encode a physicallaw

In the classical picture of a physical law, we expect thefollowing:

I The physical law should be invariant undertime-translation

I The physical law should be invariant under spatialtranslation

I The physical law should be invariant under anyorthogonal transformation (that is, anydistance-preserving transformation)

We have seen that the heat equation encodes all theseproperties.Hence, it may well encode a physical law.

The Heat Equation

Vipul Naik



Heat and its flow

Temperature and twolaws relatingtemperature to heat





Maximumprinciples











The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Heat and heat change

The heat of a body (or object) is a kind of total measure ofsome energy in it. Since heat is a kind of measure totalledacross the volume, we can talk of the thermal density or theheat density at a point, and the total heat of the body is:∫

Vh(x)

where h(x) denotes the thermal density at the point x .

Note that the total heat at a body is not something of directrelevance; what is of relevance, though, is the difference inheat across time. That is, we given times t and t ′, we areinterested in:

δt′t h(x)

that is, the difference between the heat density of spatialpoint x at times t ′ and t.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Heat and heat change

The heat of a body (or object) is a kind of total measure ofsome energy in it. Since heat is a kind of measure totalledacross the volume, we can talk of the thermal density or theheat density at a point, and the total heat of the body is:∫

Vh(x)

where h(x) denotes the thermal density at the point x .Note that the total heat at a body is not something of directrelevance; what is of relevance, though, is the difference inheat across time. That is, we given times t and t ′, we areinterested in:

δt′t h(x)

that is, the difference between the heat density of spatialpoint x at times t ′ and t.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Instantaneous picture of heat change

At any given instant of time and at any point in the body,heat is flowing in some direction, and with some magnitude.Thus, the instantaneous picture of heat flow is a vector fieldon the body that associates to each point the heat flowvector.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Heat change in a region of finite volumeHeat flowing through a body does not necessarily mean thatheat will accumulate in it. This is analogous to the fact thatwater flowing through a pipe does not cause any wateraccumulation in it.

The heat change in a body over a period of time is given by:

Heat change = Total heat flown in− Total heat flown out

Differentiating with respect to time:

Heat change per unit time = Rate of heat inflow per unit time

−Rate of heat outflow per unit time

This is with respect to a region (portion) in the body ofpositive volume.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Heat change in a region of finite volumeHeat flowing through a body does not necessarily mean thatheat will accumulate in it. This is analogous to the fact thatwater flowing through a pipe does not cause any wateraccumulation in it.The heat change in a body over a period of time is given by:

Heat change = Total heat flown in− Total heat flown out




This is with respect to a region (portion) in the body ofpositive volume.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Change in heat densityTo measure the heat density, we need to differentiate theformula:



For this, take a point x , and look at every line through x .We want to measure the outward heat flow through x , minusthe inward heat flow through x . In other words, we want tomeasure at which the rate of heat flow changes in a verysmall neighbourhood of x . This corresponds to the physicalnotion of divergence, viz:

Rate of heat density change per unit time

= ∇.Heat flow vector at the point

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Temperature controls heat flow

The temperature at a point is an intrinsic value that governsthe way heat flows through the point. Namely, thetemperature is the potential function whose gradient definesthe heat flow, or in other words, if u denotes thetemperature at position x and time t:

Heat flow vector through x at time t = k(∇u)(t, x) (1)

where k is some suitable constant that measures thermalconductivity and ∇ is the gradient function, defined as:

∇u = x 7→ (∂u

∂x1(x),

∂u

∂x2(x), . . . ,

∂u

∂xn(x))

In other words, heat flows in order to equalize temperature.The k is a conductivity constant that depends on the natureof the material.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Temperature depends on heat

The temperature at a point is related to the heat density atthe point as follows:

Heat density = Mass density× Specific heat× Temperature


Rate of heat density change = Mass density× Specific heat

×Rate of temperature change

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Putting the things together

We have the following:

I The temperature is a scalar function on the body that isproportional to the heat density at the point

I The heat flow vector is a vector function on the bodythat is proportional to the gradient of the temperaturefunction at the point

I The rate of heat change at a point is the divergence ofthe heat flow vector

Putting all these together, we get the heat equation:

∂u

∂t= k∆u

where ∆ is the Laplacian function, viz the composite of thegradient function and the divergence function.

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Notational simplification

To simplify notation, we adopt the convention of usingsubscripts for derivatives. That is, we denote ∂u/∂x as ux ,and ∂2u/∂x∂y as uxy .

With this notation, the heat equation becomes:

ut = k∆u

The Heat Equation

Vipul Naik



Heat and its flow






Maximumprinciples


Notational simplification

To simplify notation, we adopt the convention of usingsubscripts for derivatives. That is, we denote ∂u/∂x as ux ,and ∂2u/∂x∂y as uxy .With this notation, the heat equation becomes:

ut = k∆u

The Heat Equation

Vipul Naik




Recapitulation

Obvious structuralproperties

What improves andwhat is unchanged




Maximumprinciples











The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


Properties that we already saw

We saw the following about the heat equation:

I It is autonomous in t and x . That is, it is invariantunder time-translation and spatial translation

I It is invariant under orthogonal transformations in thespatial variables. This essentially follows from the factthat the Laplacian is invariant under orthogonaltransformations

We now look at some other properties that arise from themathematical structure and the physical interpretation.

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


Linearity

One thing we can say about the equation:

ut = k∆u

is that it is linear, that is:

I If u and v are two solutions, so is u + v

I If u is a solution and λ ∈ R, λu is also a solution

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


Dependence on k

We would like to know whether solutions to the heatequation for one value of k are related to solutions foranother value of k.

In fact here is an obvious relation for k 6= 0:u is a solution to ut = k∆u if and only if (t, x) 7→ u(kt, x) isa solution to ut = ∆u.Thus, we can restrict ourselves to a study of the equation:

ut = ∆u

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


Dependence on k

We would like to know whether solutions to the heatequation for one value of k are related to solutions foranother value of k.In fact here is an obvious relation for k 6= 0:u is a solution to ut = k∆u if and only if (t, x) 7→ u(kt, x) isa solution to ut = ∆u.Thus, we can restrict ourselves to a study of the equation:

ut = ∆u

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


Linear functions are solutions

The solution to the equation:

∆y = 0

is of the formy(x) = l(x) + b

where l is a linear functional and b ∈ R.Thus, the map:

u(t, x) = l(x) + b

gives a solution to the heat equation.This also tells us that the maps of the form:

u 7→ ((t, x) 7→ u(t, x) + l(x) + b)

are symmetries of the solutions to the heat equation.

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


The total heat is unchanged

The intuition should tell us that the total heat change of thesystem equals the amount of heat that flows out through theboundary.

The mathematical justification for this is as follows: the rateof heat change is the divergence of the heat flow vector field.Hence, its integral over the whole volume equals the integralover the boundary area of the heat flow vector field. This isthe area flowing through the boundary.In particular, if the object doesn’t have boundary, or if it isinsulated at the boundary, then the total heat in the systemdoesn’t change.

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


The total heat is unchanged

The intuition should tell us that the total heat change of thesystem equals the amount of heat that flows out through theboundary.The mathematical justification for this is as follows: the rateof heat change is the divergence of the heat flow vector field.Hence, its integral over the whole volume equals the integralover the boundary area of the heat flow vector field. This isthe area flowing through the boundary.In particular, if the object doesn’t have boundary, or if it isinsulated at the boundary, then the total heat in the systemdoesn’t change.

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


The extent of heat variation keeps reducing

We want to say something like: there is some quantity thatmeasures the extent to which the heat deviates from itsstandard value, such that that quantity keeps reducing asheat flows within the body.

Coming up with an exact description of this quantity is ahard task. The rough idea would be that this should be thenegative of a physical notion of entropy.

The Heat Equation

Vipul Naik




Recapitulation






Maximumprinciples


The extent of heat variation keeps reducing

We want to say something like: there is some quantity thatmeasures the extent to which the heat deviates from itsstandard value, such that that quantity keeps reducing asheat flows within the body.Coming up with an exact description of this quantity is ahard task. The rough idea would be that this should be thenegative of a physical notion of entropy.

The Heat Equation

Vipul Naik





Flow andtime-dependent vectorfield

Picture of the flow

The heat equation asa flow



Maximumprinciples











The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Differential equation without time

Suppose we have a differential operator F taking as inputfunctions y1, y2, . . . , ym of independent variablesx1, x2, . . . , xn. F could depend on the values yi as well astheir partial derivatives in the xjs. Consider the differentialequation:

F ≡ 0

That is, we want to determine choices of the functions yi

such that F (y1, y2, . . .) = 0 for all i .For the moment, we concentrate on situations with only onedependent variable y .

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Moving towards a solution in discrete time

To find a solution function y , we can try finding a map Gwhich takes as input a function y : Rn → R and outputsanother function G (y) : Rn → R, such that the fixed pointsof G are precisely the solution functions y .

The idea is to then start off with any arbitrary function y ,and compute the iterated sequence y ,G (y),G 2(y), . . ..

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Moving towards a solution in discrete time

To find a solution function y , we can try finding a map Gwhich takes as input a function y : Rn → R and outputsanother function G (y) : Rn → R, such that the fixed pointsof G are precisely the solution functions y .The idea is to then start off with any arbitrary function y ,and compute the iterated sequence y ,G (y),G 2(y), . . ..

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


In more fancy language

In more fancy language what we have done is constructed amap u : N× Rn → R satisfying the condition:

un+1(x)− un(x) = F (un, x)

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Moving towards a solution in continuous time

Recall the setup: we want to find a function y such thatF (y , x1, x2, . . . , xn) = 0 where F may also involve partialderivatives.

Consider a function u : R× Rn → Rn, written as u = ut(x)where t is the time parameter, satisfying the followingdifferential equation:

∂u

∂t= F (u, x1, x2, . . . , xn)

Now suppose y is a solution to the differential equationassociated with F . Then the function u : (t, x) 7→ y(x) (thatis, a function independent of t) is clearly a solution to thisdifferential equation.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Moving towards a solution in continuous time

Recall the setup: we want to find a function y such thatF (y , x1, x2, . . . , xn) = 0 where F may also involve partialderivatives.Consider a function u : R× Rn → Rn, written as u = ut(x)where t is the time parameter, satisfying the followingdifferential equation:

∂u

∂t= F (u, x1, x2, . . . , xn)

Now suppose y is a solution to the differential equationassociated with F . Then the function u : (t, x) 7→ y(x) (thatis, a function independent of t) is clearly a solution to thisdifferential equation.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


How this parallels the discrete case

In the discrete case, the left-hand side was:

un+1 − un

And we hope to reach some stage n (either finite or infinity)where this left-hand side vanishes (and hence we get asolution)

And in the continuous case, the left-hand side was:

∂u

∂t

And we hope to reach some time t (either finite or infinity)where this left-hand side vanishes (and hence we get asolution)

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


How this parallels the discrete case

In the discrete case, the left-hand side was:

un+1 − un

And we hope to reach some stage n (either finite or infinity)where this left-hand side vanishes (and hence we get asolution)And in the continuous case, the left-hand side was:

∂u

∂t

And we hope to reach some time t (either finite or infinity)where this left-hand side vanishes (and hence we get asolution)

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


The general picture

The picture is this:We have the set of all possible functions y : Rn → R.Now, suppose we start off with a function y0. We want toinvestigate the conditions under which we can find a solutionu to the differential equation such that y0 is the functionu(0, ). Such a solution can be viewed as follows: we startoff at y0 at time 0, and then flow along in the space ofpossible functions with time.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Some immediate questions

Here are natural properties we would seek for the flowequation:

I Short-time existence: This means that the solution uis defined for all x and for t in some neighbourhood of 0

I Global existence: This means that the solution u isdefined for all x and all t

I Uniqueness: This means that any two solutions(defined in suitable time neighbourhoods) must beequal wherever they are both defined

Short-time existence and uniqueness are typically shownusing the general theory of existence and uniqueness ofsolutions to differential equations, while global existence mayrequire further exploitation of the particular structure of thedifferential equation.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Bidirectionality of the flowOne fundamental way in which the continuous flow differsfrom the discrete flow is that it is bidirectional. For thediscrete flow, we had defined:

G (x) = x + F (x)

Now if x and G (x) were fairly close, we could possibly thinkof F (x) as being equal to F (G (x)), and get:

G (x) = x + F (G (x))

which would allow us to write:

G−1(x) = x − F (x)

Making the flow continuous actually helps us rigorize this,namely, the flow for −F is the reverse of the flow for F .(this is essentially because in the continuous thing, adjacentthings are close enough).

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


The fixed points of the flow

We know that the fixed points of the flow are precisely the ythat are solutions to F (y) = 0. We thus have the picture:

I There will be some fixed points. These are points thatdon’t move under the flow at all.

I There will be some paths defined on finite timeintervals, and some paths defined on infinite timeintervals

I If a path converges at the limit, then the point ofconvergence must be a fixed point under the flow. Forpaths defined globally, the limits at ∞ and at −∞ mayboth give (possibly distinct) fixed points

I There may be some circular flows as well – flows whichkeep going round and round

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


The flow for the Laplacian

The heat equation is the flow equation corresponding to theLaplacian, viz F = ∆ here.

Thus, we can apply all the ideas of flows to studying theheat equation.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


The flow for the Laplacian

The heat equation is the flow equation corresponding to theLaplacian, viz F = ∆ here.Thus, we can apply all the ideas of flows to studying theheat equation.

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


A definite directionality

In the case of the heat equation, there is a definitedirectionality to things. That is, if k = 1, then the heatequation evolves towards a fixed point as t →∞, ratherthan as t → −∞.In fact, there are a number of quantities we can associatewith the heat equation that:

I Are minimal (respectively maximal) when we are at afixed point

I Decrease (respectively increase) monotonically at ageneral point, with the decrease (respectively increase)becoming steadily zero only once we reach a fixed point

The Heat Equation

Vipul Naik






Picture of the flow




Maximumprinciples


Possibilities for decreasing/increasing quantities

I A bounding range for the heat function at any giventime. This is related to so-called maximum principles

I Functions that measure the average deviation of theheat function from its mean value. These include thingslike entropy functions

The Heat Equation

Vipul Naik






Separation ofvariables


Maximumprinciples











The Heat Equation

Vipul Naik








Maximumprinciples


In the one-dimensional case

For the function u = u(t, x)

∂u

∂t=

∂2u

∂x2

We try to hunt for solutions of the form:

u(t, x) = f (t)g(x)

Such solutions are termed multiplicatively separable.This technique is termed separation of variables.

The Heat Equation

Vipul Naik








Maximumprinciples


ODEs formed after we separate variables

After we separate the variable, we get the following:

f ′(t)

f (t)=

g ′′(x)

g(x)

Since this is true for every t and every x , we can set this toa value λ, and we obtain:

f ′(t) = λf (t)

And parallelly obtain:

g ′′(t) = λg(t)

The Heat Equation

Vipul Naik








Maximumprinciples


Linear combinations of multiplicatively separablesolutions

Since the heat equation is linear, any linear combination ofmultiplicatively separable solutions is also a solution.Further, any solution that is the convergent sum of aninfinite series whose terms are all multiplicatively separablesolution.

The question then would be: can every solution be obtainedin this form? That is, can the flow corresponding to everypoint be obtained as an “infinite linear combination” ofmultiplicatively separable solutions?For the heat equation, the answer in fact turns out to be yes.

The Heat Equation

Vipul Naik








Maximumprinciples



Since the heat equation is linear, any linear combination ofmultiplicatively separable solutions is also a solution.Further, any solution that is the convergent sum of aninfinite series whose terms are all multiplicatively separablesolution.The question then would be: can every solution be obtainedin this form? That is, can the flow corresponding to everypoint be obtained as an “infinite linear combination” ofmultiplicatively separable solutions?

For the heat equation, the answer in fact turns out to be yes.

The Heat Equation

Vipul Naik








Maximumprinciples



Since the heat equation is linear, any linear combination ofmultiplicatively separable solutions is also a solution.Further, any solution that is the convergent sum of aninfinite series whose terms are all multiplicatively separablesolution.The question then would be: can every solution be obtainedin this form? That is, can the flow corresponding to everypoint be obtained as an “infinite linear combination” ofmultiplicatively separable solutions?For the heat equation, the answer in fact turns out to be yes.

The Heat Equation

Vipul Naik








Maximumprinciples


The more general situation of a flowThe same idea of separation of variables that we used for theheat equation works in the greater generality of a flow. Infact, it works well when the F for which we are consideringthe differential equation, is a linear differential operator.Consider:

∂u

∂t= F (u)

Expressing u(t, x) as f (t)g(x) we obtain the system ofdifferential equations:

f ′(t) = λf (t)

F (g)(t) = λg(t)

Thus, g must be an eigensolution for F with eigenvalue λ.We again have that under suitable assumptions, everysolution is an infinite linear combination of such separablesolutions.

The Heat Equation

Vipul Naik







The heat equation asa diffusion equation

Gradient and reactionterms

Supersolutions andsubsolutions

Maximumprinciples











The Heat Equation

Vipul Naik










Maximumprinciples


The more general idea of diffusion

Instead of thinking of the heat equation in terms of heatcontent and temperature, we can view it in terms of anuneven mass density of a material in a body. If the massdensity is not equal everywhere, there is a tendency for massto flow from higher density to lower density, resulting ingreater mass equalization.We can deduce, using similar reasoning to that for the heatequation, that if u denotes the mass density at a point, usatisfies the heat equation.Hence, the heat equation is also termed the diffusionequation.

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Maximumprinciples


Gradient term and reaction term

The heat equation with gradient term is given by thefollowing general equation:

ut = ∆u + 〈X ,∇u〉

The heat equation with both gradient term and scalar term(also called reaction term) is given by the following generalequation:

∂u

∂t= ∆u + 〈X ,∇u〉+ H(u)

If H(u) = βu we say that we have a heat equation withlinear reaction term.A general equation of the above setup is termed areaction-diffusion equation.

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Maximumprinciples




ut = ∆u + 〈X ,∇u〉


∂u

∂t= ∆u + 〈X ,∇u〉+ H(u)

If H(u) = βu we say that we have a heat equation withlinear reaction term.

A general equation of the above setup is termed areaction-diffusion equation.

The Heat Equation

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Maximumprinciples




ut = ∆u + 〈X ,∇u〉


∂u

∂t= ∆u + 〈X ,∇u〉+ H(u)

If H(u) = βu we say that we have a heat equation withlinear reaction term.A general equation of the above setup is termed areaction-diffusion equation.

The Heat Equation

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Maximumprinciples


Physical significance of the reaction term

Reaction terms arise in the physical situation as terms thatalter the heat at a point without heat flowing to or from theneighbouring points. The word “reaction” stems, forinstance, from the fact that when heat is flowing through amaterial, too much of it may concentrate at a point, leadingto some chemical change at that point.

Note that once there are reaction terms, the total heat ofthe system is no longer kept constant.

The Heat Equation

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Maximumprinciples


Physical significance of the reaction term

Reaction terms arise in the physical situation as terms thatalter the heat at a point without heat flowing to or from theneighbouring points. The word “reaction” stems, forinstance, from the fact that when heat is flowing through amaterial, too much of it may concentrate at a point, leadingto some chemical change at that point.Note that once there are reaction terms, the total heat ofthe system is no longer kept constant.

The Heat Equation

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Maximumprinciples


The four kinds of reaction terms

I Positive and positively related reaction terms: These arereaction terms that are positive in sign and arepositively related to the heat function. This means thatthe greater the heat density at a point, the more it rises.

I Positive and negatively related reaction terms: Theseare reaction terms that are positive in sign and arenegatively related to the heat function. This means thatthe greater the heat density at a point, the less it rises.

I Negative and positively related reaction terms

I Negative and negatively related reaction terms

The Heat Equation

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Maximumprinciples


All these heat equations satisfy the flow model

The general form of heat equation with both gradient andreaction term is also a flow equation, corresponding to Fbeing:

y 7→ ∆u + 〈X ,∇u〉+ H(u)

Hence, we can view them in the same way (doing the samekind of analysis of flows) as we did for the heat equation.However, if the reaction term has a time-dependence (thatis, it is dependent on t), then the heat equation cannot beviewed as a flow equation corresponding to a differentialoperator.

The Heat Equation

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Maximumprinciples


Supersolutions and subsolutions

u is termed a supersolution at (t, x) if:

∂u

∂t≥ ∆(u) + 〈X ,∇u〉+ F (u)

u is termed a supersolution if it is a supersolution for all xand all tSimilarly, u is a subsolution at (t, x) if:

∂u

∂t≤ ∆(u) + 〈X ,∇u〉+ F (u)

u is termed a subsolution if it is a subsolution for all x andall t

The Heat Equation

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Maximumprinciples

Maximum principlefor the heat equation

Effect of a reactionterm











The Heat Equation

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Maximumprinciples




The statement for the heat equation

The following is the statement for the heat equation:Let u be a solution to the heat equation with the propertythat u(0, x) ∈ [T1,T2] for all x ∈ Rn. Then, the maximumprinciple states that:

u(t, x) ∈ [T1,T2] ∀x ∈ Rn,∀t ≥ 0

In other words, whatever bounds/limits control the heatfunction at the initial time, control the heat function at alllater times.

By the invariance under time-translation, we can say that therange of u(t, x) for given t, keeps reducing as t increases.

The Heat Equation

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Maximumprinciples




The statement for the heat equation

The following is the statement for the heat equation:Let u be a solution to the heat equation with the propertythat u(0, x) ∈ [T1,T2] for all x ∈ Rn. Then, the maximumprinciple states that:

u(t, x) ∈ [T1,T2] ∀x ∈ Rn,∀t ≥ 0

In other words, whatever bounds/limits control the heatfunction at the initial time, control the heat function at alllater times.By the invariance under time-translation, we can say that therange of u(t, x) for given t, keeps reducing as t increases.

The Heat Equation

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Maximumprinciples




For a more general reaction-diffusion equation

A general reaction-diffusion equation is said to satisfy theweak maximum principle if for any solution u of thatequation:

u(t, x) ∈ [T1,T2] ∀x ∈ Rn,∀t ≥ 0

In other words, whatever bounds/limits control the heatfunction at the initial time, control the heat function at alllater times.

The Heat Equation

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Maximumprinciples




A second version of the maximum principle

Here’s another version of the maximum principle. Considerthe equation:

∂u

∂t= ∆u + 〈X ,∇u〉

Then suppose we have an α ∈ R satisfying the following twoconditions:

I u(x , 0) ≥ α

I u is a supersolution of the heat equation at any(t, x) ∈ [0,T )× Rn for which u(t, x) < α.

Then u(t, x) ≥ α for all (t, x) ∈ [0,T )× Rn.

The Heat Equation

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Maximumprinciples




Reaction terms can be disruptive

Consider a positive and positively related reaction term.What this does is to push up the heat at various points.Moreover, those points where the heat density is higher,tend to get pushed up more. Thus, in this case, the reactionterm actually increases disparity and obstructs the process ofattaining equilibrium.We want to prove that a maximum principle persists even inthe presence of reaction terms.

The Heat Equation

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Maximumprinciples


In the physical world

In the mathematicalworld










The Heat Equation

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Maximumprinciples




For physical situations

The heat equation that we studied originally is a prototypefor general reaction-diffusion equations, which are used tomodel many physical and chemical situations. Further, manyof the concepts such as energy, entropy, maximum principleand so on, that we develop for the heat equation, motivatecorresponding notions for more general reaction-diffusionequations.

The heat equation also inspires ideas in more general flowequations governed by laws qualitatively very different fromthose for the heat equation.

The Heat Equation

Vipul Naik







Maximumprinciples




For physical situations

The heat equation that we studied originally is a prototypefor general reaction-diffusion equations, which are used tomodel many physical and chemical situations. Further, manyof the concepts such as energy, entropy, maximum principleand so on, that we develop for the heat equation, motivatecorresponding notions for more general reaction-diffusionequations.The heat equation also inspires ideas in more general flowequations governed by laws qualitatively very different fromthose for the heat equation.

The Heat Equation

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Maximumprinciples




The leading ideasThe heat equation, which originally started out from thephysical motivation, has helped in the development of awhole lot of mathematics. To see how, let’s single out theimportant aspects of flow equation:

I We have a differential operator F and are looking atfunctions annihilated by F .

I We consider the flow equation ut = F (u) and considerthe way arbitrary initial solutions evolve under this flowequation. If we can find some initial solutions for whichthe flow converges at ∞, we get functions annihilatedby F .

I We develop certain special techniques, such as the useof energy, entropy, and maximum principles, to showthat the flow actually comes closer and closer tosomething, viz it converges.

The above gives a concrete recipe for trying to locatefunctions annihilated by F .

The Heat Equation

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Maximumprinciples




Ricci flows and the Hamilton program

Hamilton formulated a program to solve the famousPoincare conjecture using a technique called Ricci flows.The idea (rough sketch) was:

I We consider a differential operator on the space ofRiemannian metrics on a differential manifold (the Riccicurvature tensor), such that we are interested in thosemetrics which are annihilated by this operator

I We consider the flow equation associated with thisoperator – the so-called Ricci flow

I We then try to find some flows which actually convergeand hence find metrics annihilated by the Riccicurvature tensor

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