UA MATH 456 - Conservation and dissipation principles for PDE models

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Conservation and dissipation principles for PDEmodels1 Conservation lawsThe notion of conservation - of number, energy, mass, momentum - is afundamental principle that can be used to derive many familiar partial dif-ferential equations.Let u(x, t) be the density of any quantity - heat, momentum, probability,bacteria, etc. The amount of u in a (say 3-dimensional) region Ω isZΩu dV.To say that a quantity is conserved means that it is only gained or lostthrough either (1) domain boundaries or (2) because of sources and sinksin the domain. The flow or flux, of u(x, t) can be thought of as a vector fieldJ(x, t). This is defined so that J·ˆndA is the amount of u flowing in directionˆn per unit time across a small area dA. If Q(x, t) is the rate of inflow of u(outflow if negative) then the conservation of u on any domain Ω impliesddtZΩu dV =ZΩ∂u∂tdV = −Z∂ΩJ · ˆndA +ZΩQ(x, t)dV. (1)Here ∂Ω denotes the boundary of Ω and ˆn is the outward normal. Equation(1) says that the change in u is due to flow out of Ω and source terms in Ω.The surface flux integral in (1) can be converted to a volume integral byusing the divergence theorem, givingZΩ∂u∂t+ ∇·J − QdV = 0.The important point is that this is true for every region Ω, which means thatthe integrand must be exactly zero. This gives the differential equation∂u∂t+ ∇·J = Q. (2)(in some contexts, this is called the transport or continuity equation). Thisequation is valid for any spatial dimension (divergence in one dimensionis just the regular derivative ∂/∂x). The flux and source terms are problemspecific. Some examples are given below.11.1 Boundary ConditionsIf a quantity is conserved, it still may be gained or lost by flowing throughthe boundaries of the physical domain. This implies that the flux J must bespecified on the domain boundary. For example, in the diffusion equation,the flux is −D∇u, so a valid boundary condition could be written−D∇u · ˆn = J0(x), x ∈ ∂Ω,where ˆn is the outward normal to the domain boundary ∂Ω. Often theboundaries are called “no-flux” or “insulating” (in the case of heat diffu-sion), which leads to the very common “Neumann” boundary condition∇u · ˆn = 0.Note that other types of boundary conditions can be used in conserva-tion laws. For example, if the temperature is fixed on the boundary in heatdiffusion, this corresponds to a “Dirichlet”-type boundary conditionu = U(x), x ∈ ∂Ωinstead. In this case, the flux across the boundary is generally not zero,but is simply whatever flux is needed to maintain the Dirichlet boundarycondition.1.2 ExamplesSimple transport in one space dimension. Suppose that u = u(x, t) istransported at velocity c. This means that the scalar flux is J = cu. Withoutsource terms, (2) becomesut+ cux= 0. (3)Traffic flow. Unlike simple transport, automobiles do not generally travelat a uniform speed. The simplest way of modeling this situation is takingthe speed c = c(u) as a decreasing function of the density of cars u. Thismakes sense most of the time: greater densities lead to slower speeds. Thenthe scalar flux is J = c(u) and (2) becomesut+ (c(u)u)x= 0. (4)This has the form of a one-dimensional hyperbolic conservation law ut+f(u)x= 0. The choice f = u2/2 gives rise to what is known as Burger’sequation.2Diffusion. The idea of a quantity diffusing means that its flux has a direc-tion toward regions of less density. Mathematically, this can be modeled asJ = −D∇u since the gradient points in the direction of greatest increase.(The constant D is known as the diffusivity, and is measured in units oflength squared per unit time). This is known as Fick’s law (or Fourier’slaw if u is heat). In this case, (2) becomesut= D∇·∇u = D∆u, (5)which is known as the diffusion equation.Diffusion with a nonlinear source. Suppose that the source term Q(x, t)is a function f (u) of u itself. This might be the case for a chemical reaction(where Q is reaction rate) or biological reproduction (for example, Q =u(1 − u) as in the logistic equation). Then one obtainsut= ∆u + f(u), (6)which is a simple example of a reaction-diffusion equation.Chemotaxis. Various cells and microorganisms move in response to chem-ical gradients. For example, certain bacteria are drawn toward oxygen, andwhite blood cells may move in response to chemicals produced at the siteof an infection.Let u(x, t) be the time-dependent density of cells, and let c(x, t) be thedensity of chemoattractant concentration. The main idea is that the flux ofcells is in the same direction as the (spatial) gradient of c - in other words,the cells seek a direction where there is the greatest concentration. Themagnitude of the flux is, on the other hand, typically a fixed number M,which has to do with the mobility of cells. It follows that the flux of cellsdue to chemotaxis isJc= M∇c|∇c|.In real situations, one also has standard diffusion of cells modeled as thediffusive flux Jd= −D∇u. The equations of motion are a combination ofthese fluxes:ut= −∇·(Jc+ Jd) = −M∇·∇c|∇c|+ D∆u. (7)Of course, one can also consider the evolution of chemoattractant by diffu-sion. If the organism produces the chemoattractant itself (as in the famousslime mold example), one arrives at a coupled system for u and c known asthe Keller-Segel equations.3Wave equation. This example is different from the rest. In this case, theconserved quantity is momentum, which is the time derivative of the dis-placement u (properly speaking, one needs to multiply by mass density).Momentum flux occurs because forces (if the wave arises from, say, an elas-tic body) are transmitted spatially, and are proportional to −∇u. With aconstant of proportionality equal to one, the fact that momentum is con-served means(ut)t− ∇ · ∇u = 0which is the wave equation with unit speed.1.3 Conserved and dissipated quantitiesIf u(x, t) is some function, a mapping from u to the real numbers is called afunctional. For example,ZΩudx,ZΩu2xdx,ZΩu4xxdxare all examples of functionals. It often happens that functionals representquantizes of physical interest – mass, energy, entropy. They can also havemathematical usefulness in their own right, as they may give knowledgeof “coarse-grained” quantities of the unknown solution u(x, t).Suppose F is some functional of u(x, t) of the formF [u] =ZΩf(u, ux, ...)dx.so that F depends on t, but not on the variable x


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