Partial Differential Equations of Parabolic Type.
A transfinite type theory with type variables .
Some of recent developments, including recent results, ideas, techniques. P^j and approaches, in the study of degenerate partial differential equations are surveyed. Several examples of nonlinear degenerate, even mixed, partial differen- .
The solution to these funda- . Our emphasis is on exploring and/or developing unified math- . The potential approaches. Qh ' techniques. We remark that most of the important problems for nonlinear degenerate. Introduction. start with several important examples of degenerate/mixed linear degenerate equations.
Then we present several examples of nonlinear degen- . These examples in- . The potential approaches we have identified and/or devel- .
Full text of 'On Degenerate Partial Differential Equations'. DIFFERENTIAL EQUATIONS 3 and parabolic. Partial Differential Equations of Parabolic Type. Partial Differential Equations of Parabolic Type Avner Friedman. Language: en Download (djvu. A First Course in Partial Differential Equations with complex variables and transform methods. Numerical Methods for Elliptic and Parabolic Partial.
The solution to these. Date: May 1. 8, 2. Primary: 3. 5- 0. L6. 5, 3. 5J7. 0, 3.
K6. 5, 3. 5L8. 0, 3. M1. 0, 3. 5L4. 0. L4. 5, 3. 5K1. 0,3.
Partial Differential Equations of Mathematical Physics was developed chiefly with. Partial Differential Equations.
K1. 5, 3. 5J1. 5, 3. Q0. 5, 7. 6G2. 5, 7. H0. 5, 7. 6J2. 0, 7.
N1. 0, 7. 6S0. 5, 7. L0. 5, 5. 3C4. 2, 5. C2. 0, 3. 5M2. 0. Partial differential equations, degenerate, mixed, elliptic, hyperbolic, para- . Navier- Stokes equations, Euler equations, kinetic. CHEN. degenerate partial differential equations. On the other hand, most of the important prob- .
With these achievements, it is the time to revisit and attack nonlinear degenerate. In Section 2, we present several important. In Section 3, we first reveal a natural connection between degenerate hyperbolic. Euler- Poisson- Darboux equation through the en- .
Young measure, and then we discuss how this connection can be applied to. In Section 6, we discuss how the singular limits to. Navier- Stokes equations to the isentropic Euler equations. Linear Degenerate Equations. In this section, we present several important examples of linear degenerate, even mixed. Consider the partial differential equations of second- order.
C R d , (2. 1). where ciy (x) , fy (x) , c(x) , and /(x) are bounded for x . In particular, when. The simplest example is. A(t,x)=(. That. is, for the linear case, the matrix (ajj(t, x)) or (ajj(x)) is not positive definite or even. A(t, x) are not distinct. Degenerate Equations and Mixed Equations. Two prototypes of linear degen- .
Euler- Poisson- Darboux equation. Interrelations between the Linear Equations. The above linear equations are.
Seek. spherically symmetric solutions of the wave equation (I2. P . v(t,r) = n(t,x), r = . The Tricomi Equation (.
Under the coordinate transformations. The study of nonlinear partial differential equations has been focused mainly.
The three different types of. Great progress has been made through the efforts of several generations of mathe- .
The examples include nonlinear degenerate hyperbolic systems. Euler. equations for compressible flow in fluid mechanics, and the Gauss- Codazzi system for iso- . Such degenerate, or mixed, equations naturally. The solution to these. Nonlinear Degenerate Hyperbolic Systems of Conservation Laws. Nonlinear hyperbolic systems of conservation laws in one- dimension take the following.
U + d x F(U) = 0, U. If) is equivalent to. U + VF(U)d x U = 0, U e R n , (t, x) e R 2 + . CHEN. as the degenerate set. If the set T> is empty, then this system is strictly hyperbolic and. Such a set allows a degree of interaction, or nonlinear. On the other hand, degenerate hyperbolicity of sys- .
For example, for three- dimensional hyperbolic systems. Lax . The result is also true when the systems have . Then the plane wave solutions of such systems are governed by the. T> ^ 0. This situation reflects in part the physical. For this reason. attention is focused on solutions in the space of discontinuous functions, where one can.
The key issue is whether the approximate. Solving this issue involves two aspects: one is to construct good approximate solu- .
Connection with the Euler- Poisson- Darboux Equation: Entropy and Young. Measure. The connection between degenerate hyperbolic systems of conservation laws. Euler- Poisson- Darboux equation is through the entropy. Clearly, any C 1 solution satisfies.
For n = 2, the Frobenius condition always holds. The entropy space is infinite- dimensional and is represented by two families. However, for the case T> ^ 0, the situation is much more. A . The typical form of such equations is. Wl - w 2 ) P(^,wi- w 2 ). U! 1 w. 2 r l^ d wi r. For an arbitrary sequence of measurable maps.
L. In particular, U e (t,x) converges strongly to U(t,x) if. Young measure Vt,x is equal to a Dirac mass concentrated at U(t, x) for. This indicates that proving the compactness of. U e (t,x) is equivalent to solving the functional equation (. If one clarifies the structure of the Young measure f t)X (A), one can. For example, if one can.
Young measures Uf j. X , (t, x) G M. 2 ,, are Dirac masses, then one can. U e (t,x) almost everywhere. One of the principal difficulties for. This is due to possible singularities of. The larger the set of C 2. Hyperbolic Conservation Laws with Parabolic Degeneracy.
One of the pro- . Euler equations with the form. P )) = o, . For p > 0, v = m/p. The physical region for (I3. P is . CHEN. In terms of the variables (p,v), r\ is determined by the following second- order linear. V - k'(p) 2 d vv ri = 0. For example, the mechanical energy.
V* = o . Then the family of weak entropy functions is described by. P,v)= / x(p,v,s)ip(s)ds, (3. Jm. By construction, rj(0,v) = 0, rj p (0,v) = ip(v).
One. can prove that, for < p < C, . Large- data theorems have also been obtained for general 7 > 1 in. The difficult point in bounding. This difficulty is a reflection of the fact that the. D/0. The. existence problem for usual gases with general values 7 G (1, .
The case 7 > 3 was treated by Lions- Perthame- Tadmor. Lions- Perthame- Souganidis . A compactness framework has been established even for the.
Chen- Le. Floch . The solutions under consideration will remain. This. means that the pressure law p(p) has the same singularity as ^7=1 K j P lj near the vacuum. Consider system ()3. Assume that a sequence of functions (p . Moreover, there exists a global en- . Cauchy problem (.
Furthermore, the. For the Euler equations, to obtain. Young measure V(t, x ) ls a Dirac mass in the (p, m)- plane, it suffices to prove that.
V(t,x)i i s either a single point or a subset. Dirac measure. CHEN. It is used that (.
The new idea of applying the technique of fractional derivatives. Chen . Motivated by a. Here. cr* is the entropy- flux kernel defined as.
The main issue is to construct all of the weak entropy pairs of (I3. P . The proof in. Chen- Le. Floch . New properties. of cancellation of singularities in combinations of the entropy kernel and the associated. In particular, a new multiple- term expansion has.
Bessel functions with suitable exponents, and the optimal. The results cover, as a special example, the density- pressure law p(p) = K\ p 1. K2 p 7. 2 where 7. The proof of the. Young measure has also further simplified the proof known for the. Hyperbolic Conservation Laws with Hyperbolic Degeneracy. One of the. prototypes of hyperbolic conservation laws with hyperbolic degeneracy is the gradient.
U + d x . For any smooth nonlinear flux function. Taylor expansion about the isolated umibilic point. The first three terms including. DEGENERATE PARTIAL DIFFERENTIAL EQUATIONS 1.
The hyperbolic degeneracy enables us to eliminate the linear term by a. Such a polynomial flux contains some inessential scaling parameters. There is a. nonsingular linear coordinate transformation to transform the above system into (I3. From the viewpoint of group theory, such a reduction from. For the six dimensional space of quadratic mappings acted.
GL. Two new. types of shock waves, the overcompressive shock and the undercompressive shock, were. The overcompressive. Lax entropy condition .
It is known that. Stability of such traveling waves for the. The global existence of entropy solutions to the Cauchy problem for a special. Kan . A. different proof was given independently to the same problem by Lu . Under this framework, any approximate solution sequence, which. L. This means that the.
Riemann invariant sequence. Again, one of the principal difficulties associated with such. This is due to possible singularities of. The analysis leading to. In the first step, we have constructed regular entropy functions governed by a highly. Euler- Poisson- Darboux type equation. The first is that the coefficients of the entropy equation are.
Riemann invariant coordinates. This analysis involves a study of a corresponding. Euler- Poisson- Darboux equation and requires very complicated estimates and calculations. CHEN. An appropriate choice of Goursat data leads to the cancellation of singularities and the. Riemann invariant coordinates.
The second diffi- . Riemann invariant coordinates is, in general, irregular. A regular entropy function in the. Riemann invariant coordinates is usually no longer regular in the physical coordinates.
This has. been achieved by a delicate use of Serre's technique . Some corresponding existence theorems of global entropy solutions. The compactness has been achieved by reducing. Young measures to a Dirac mass in the physical space.
Nonlinear Degenerate Parabolic- Hyperbolic Equations. One of the most important examples of nonlinear degenerate parabolic- hyperbolic equa- . V- f(. Equation (J4.
J) is degen- . erate on the level set . Although equation (J4. J) is of parabolic nature, the solutions exhibit certain. One striking family of solutions is. Barenblatt's solutions found in . The dynamic boundary of.
For any constant states u such that. Although u(t,x) is discontinuous, . A discontinuous profile connecting the hyperbolic phase from. The well- posedness issue for the Cauchy problem is relatively well understood if one re- .
V- (A(u)Vtt), thereby obtaining a scalar hyperbolic conservation. Lax . It is equally well understood for the diffusion- . CHEN. points with certain order of degeneracy; see Brezis- Crandall . For the isotropic diffusion, Oij(u) = 0, i ^ j, some stability results. BV solutions by Volpert- Hudjaev . This approach is motivated by the macroscopic closure procedure of the.