On determinism and well-posedness in multiple time dimensions

If the evolution mapping y1 → (u, ∂y1 u) is in C^1(R_y1 : H^1 × H^0), then the energy is conserved along a solution u(·, y1, ·).

“Theorem 2.1. Suppose that the evolution mapping y 1 → ( u ( x,y 1 ,y ′ ) ∂ y 1 u ( x,y 1 ,y ′ ) ) is in C 1 ( R y 1 : H 1 × H 0 ) . Then the energy is conserved along a solution u ( · , y 1 , · ) :”

Solutions can be singular for standard classes of data.

“Indeed, we show that solutions can be singular for standard classes of data.”

For initial data on a lower-dimensional hypersurface M of mixed signature, it is not always possible to extend arbitrary data w=(w0,wα) on M to N so that the well-posedness constraint is satisfied, due to analogous lower-dimensional constraints on M.

“This is not always possible for arbitrary data w = ( w 0 , w α ) posed on M , due to analogous lower dimensional constraints on M .”

Initial data consisting of the values of the solution u(x,y) and its first normal derivatives on M, through the extension procedure, give rise to constraint-satisfying Cauchy data on a codimension-one hypersurface N.

“In section 3 we showed that initial data consisting of the values of the solution u ( x, y ) and its first normal derivatives on M , through a procedure of extension, give rise to constraint-satisfying Cauchy data on a codimension-one hypersurface N .”

In case d1=p1, it is not possible to extend arbitrary data to a function u=E(w) satisfying the constraint supp(û(ξ,η'))⊆R1.

“In case d 1 = p 1 , it is not possible to extend arbitrary data to a function u = E ( w ) which satisfies the constraint supp(ˆ u ( ξ, η ′ )) ⊆ R 1 .”

For such mixed data, it is possible to extend the data, along with attendant Sobolev bounds on the extended functions, in most cases.

“But it is possible, along with attendant Sobolev bounds on the extended functions, in most cases.”

Our work provides a rigorous analytic alternative for Woodhouse's solution method, which is not restricted to this choice of space and time dimensions.

“Our work provides a rigorous analytic alternative for his solution method, which is not restricted to this choice of space and time dimensions.”

Imposing a constraint on the initial data makes the solution process well defined.

“On the other hand, imposing a constraint on the initial data, the solution process is well defined.”

Initial value problems with data posed on a hypersurface of codimension greater than or equal to two are generally ill-posed.

“Therefore one might consider the initial value problem with data posed on a hypersurface of codimension greater than or equal to two. Such problems are generally ill-posed.”

The initial value problem on higher codimension hypersurfaces is ill-posed due to failure of uniqueness, at least when specifying a finite number of derivatives of the data.

“In contrast, we show that the initial value problem on higher codimension hypersurfaces is ill-posed due to failure of uniqueness, at least when specifying a finite number of derivatives of the data.”

The extensions from initial data on M to constraint-satisfying Cauchy data on N are highly nonunique.

“These extensions are highly nonunique, and therefore so are the resulting global solutions.”

The paper gives a generalization of Courant's theorem which extends to a broader class of equations.

“We give a generalization of Courant's theorem which extends to a broader class of equations.”

For initial value problems of type (i) and (ii) on higher codimension hypersurfaces, solutions are highly nonunique, even among H^m smooth solutions and with the constraint from section 2 imposed.

“In section 3 we consider the initial value problem for data given on higher codimension hypersurfaces, and we find that solutions are highly nonunique for the initial value problems of type (i) and (ii) above, even among H m smooth solutions and with the imposition of the constraint given in section 2.”

Using the extension operator, one can generate constraint-satisfying Cauchy data on N from initial data on M.

“Using the extension operator, we generate constraint-satisfying Cauchy data on N from initial data on M as follows:”

The solutions in X_S, X_U, and X_C are continuous with respect to their Cauchy data taken in the respective subspaces.

“These solutions are continuous with respect to their Cauchy data taken in the respective subspaces. This is the result of the next theorem.”

One may specify an arbitrary finite number of normal derivatives of the solution on the higher codimension hypersurface, and insist upon smooth solutions, yet still fail to achieve uniqueness.

“Indeed, one may specify an arbitrary finite number of normal derivatives of the solution on the higher codimension hypersurface, and insist upon smooth solutions, yet still fail to achieve uniqueness.”

The two conditions required for the kernel to satisfy the constraint E(w)∈X_C for arbitrary data w are always satisfiable, except when ξ''={0}, meaning d1=p1 and the extension subspace {(ξ'',η'')} is purely timelike.

“These two conditions are always satisfiable, except in the case ξ ′′ = { 0 } , meaning that d 1 = p 1 and the extension subspace { ( ξ ′′ , η ′′ ) } is purely timelike.”

The constructed Cauchy data satisfies the constraint, that is, (u0,u1)∈X_C, the center manifold.

“By design, this Cauchy data satisfies the constraint, that is, ( u 0 , u 1 ) ∈ X C , the center manifold.”

If d1 > p1, then there is a choice of kernel χ (indeed many) such that u=E(w) satisfies the constraint and higher Sobolev norms of u=E(w) are bounded as stated.

“Theorem 3.6. If d 1 > p 1 then there is a choice of kernel χ (indeed there are many such choices) such that u = E ( w ) satisfies”

• Also stated in: § ( c ) The extension problem for mixed spacelike and timelike data · p. 15 · plain

  “Higher Sobolev norms of u = E ( w ) are bounded as follows”

The issue of stability and uniqueness for the Cauchy problem can be addressed by imposing nonlocal constraints that arise naturally from the field equations.

“We find that the issue of stability and uniqueness for the Cauchy problem can be addressed by imposing nonlocal constraints that arise naturally from the field equations.”

A satisfactory kernel of an extension operator E can be constructed such that all functions u=E(w) in its range have Fourier support satisfying supp(û) ≤ { |η'|^2 < |ξ|^2 }.

“In doing so, we arrive at a satisfactory kernel of an extension operator E with the property that all functions u = E ( w ) in its range have Fourier support satisfying supp(ˆ u ) ≤ { | η ′ | 2 &lt; | ξ | 2 } .”

Among smooth solutions, data in an arbitrarily small ellipsoidal neighborhood of a disk in M uniquely determine the data in the envelope of its light cones.

“We do show in section 4 that among smooth solutions, data in an arbitrarily small ellipsoidal neighborhood of a disk in M uniquely determine the data in the envelope of its light cones.”

The result that the Cauchy problem is ill-posed in general but well-posed under a nonlocal constraint also applies to the wave equation with Cauchy data on a mixed signature hypersurface.

“This applies as well to the wave equation with Cauchy data on a mixed signature hypersurface.”

In the case of the ordinary wave equation (d1 = 1), solutions in X_C correspond to the full energy space H^1 × L^2.

“In the case of the ordinary wave equation ( d 1 = 1), solutions in X C correspond to the full energy space H 1 × L 2 .”

A sufficiently large class of data (w0, w(α'',0), w(0,β''), wβ1) on M extends to Cauchy data on N that is both of finite energy and satisfies the constraint.

“Finally, we show that a sufficiently large class of data ( w 0 , w ( α ′′ , 0) , w (0 ,β ′′ ) , w β 1 ) on M extends to Cauchy data on the hypersurface N which is both of finite energy and satisfies the constraint.”

Data in region R2 would lead to an ill-posed initial value problem if M were considered itself as a codimension-one hypersurface.

“where we note that the region ˜ R 2 contains the data which would lead to an ill-posed initial value problem if M were considered itself as a codimension one hypersurface.”

• Also stated in: § ( c ) The extension problem for mixed spacelike and timelike data · p. 16 · hedged

  “By itself, this region would give rise to data in L 2 ( M ) for which the Cauchy problem of mixed type is ill-posed.”

The Cauchy problem for the ultrahyperbolic equation (1.2) is ill-posed in general.

“In section 2 we use Fourier methods to show that the Cauchy problem for the ultrahyperbolic equation (1.2) is ill-posed in general but well-posed on Sobolev spaces H m if an explicit nonlocal constraint is imposed upon the Cauchy data.”

The values of a solution in an arbitrarily small neighborhood of the initial hypersurface are sufficient to determine the solution uniquely.

“In contrast to this, we conclude with a result that essentially recovers and generalizes a theorem of Courant, which shows that the values of a solution in an arbitrarily small neighborhood of the initial hypersurface are sufficient to determine the solution uniquely.”

The extension operator E is bounded on the function spaces specified in Theorem 3.4, with m the exponent of Sobolev regularity.

“Theorem 3.4. The extension operator E is bounded on the following function spaces:”

Given k, there exist constraint-satisfying data u0,u1 on N which vanish to order k on M.

“Theorem 4.1. Given k , there exist constraint-satisfying data u 0 , u 1 on N which vanish to order k on M .”

As long as d1>p1, extensions can be found for data in region R2 such that the well-posedness constraint is satisfied, by making up the default caused by | ̃ξ|<| ̃η| with a choice of large |ξ''|.

“That is, as long as d 1 &gt; p 1 , so that { ξ ′′ } is not restricted to the zero-dimensional vector space, extensions can be found in a way that the default in satisfying the constraint caused by the fact that ∣ ∣ ∣ ˜ ξ ∣ ∣ ∣ &lt; | ˜ η | can be made up with a choice of large | ξ ′′ | .”

Given w0 ∈ ̇H(M) and wα ∈ ̇H(M), the energy norm of the extension is finite.

“Theorem 3.5. Given w 0 ∈ ˙ H ( M ) and w α ∈ ˙ H ( M ) , the energy norm of the extension”

• Also stated in: § ( b ) The extension problem for general spacelike data · p. 13 · plain

  “is finite; indeed”

General initial data in H^{m+1} × H^m do not even give rise to solutions which are tempered distributions for any nonzero y1.

“However, it is clear that general initial data in H m +1 × H m do not even give rise to solutions which are tempered distributions for any nonzero y 1 .”

Even imposing the constraint discussed in section 2, which is the requirement of global existence, smooth solutions are highly non-unique.

“Moreover, even imposing the constraint discussed in section 2, which is the requirement of global existence, smooth solutions are highly non-unique.”

Specifying any finite number of derivatives of u on M does not remedy the nonuniqueness.

“In fact, if one specifies any finite number of derivatives of u on M it does not.”

For any w=(w1,w') with |w|≤1 (the unit sphere in R^{d2}), the region between the cone Rε(w) and the ellipsoid Zε is a region of determinacy for the ultrahyperbolic equation.

“We will show that for any w = ( w 1 , w ′ ) with | w | ≤ 1 (namely the unit sphere in R d 2 ), the region between the cone R ε ( w ) and the ellipsoid Z ε is a region of determinacy for the ultrahyperbolic equation.”

The resulting global solutions are nonunique.

“These extensions are highly nonunique, and therefore so are the resulting global solutions.”

Any other solution v(x,y) which takes on specified data on M up to k-many derivatives may be changed by adding this solution u to it, without changing its initial data.

“Hence any other solution v ( x, y ) which takes on specified data on M up to k -many derivatives may be changed by adding this solution u to it, without changing its initial data.”

Some constraint is necessary, as evident from the Asgeirsson mean value theorem and its consequences, as discussed in Courant (1962).

“The fact that some constraint is necessary is indeed evident from the Asgeirsson mean value theorem, and its consequences, as discussed in Courant (1962).”

• cites Courant (1962)Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

A uniqueness result for data given in an arbitrary neighborhood of such initial hypersurfaces follows from Asgeirsson's mean value theorem, as deduced by Courant.

“This failure is in contrast to a uniqueness result for data given in an arbitrary neighborhood of such initial hypersurfaces, which Courant deduces from Asgeirsson's mean value theorem.”

• cites CourantNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Therefore, there exists a globally defined solution u(x,y) which has initial data u(x,y)=u0, ∂y1 u(x,y)=u1 on N, and which vanishes to order k on M.

“Therefore, there exists a globally defined solution u ( x, y ) which has initial data u ( x, y ) = u 0 , ∂ y 1 u ( x, y ) = u 1 on N , which vanishes to order k on M .”

The existence of the extension for mixed data follows as in Theorems 3.4 and 3.5 from the construction of an extension operator E with certain boundedness properties on appropriate Sobolev spaces.

“The existence of such an extension follows as in Theorems 3.4 and 3.5 from the construction of an extension operator E with certain boundedness properties on appropriate Sobolev spaces.”

For a given Rε(w), the Holmgren-John theorem is based upon the construction of an analytic family of noncharacteristic hypersurfaces Sλ with which to sweep the region between Zε and Rε(w).

“For a given R ε ( w ), the Holmgren-John theorem is based upon the construction of an analytic family of noncharacteristic hypersurfaces S λ with which to sweep the region between Z ε and R ε ( w ).”

• cites John 1982Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Knowledge of the data in a small cylinder determines the values of all derivatives on N={ (x,y): y=0 } (if the data are smooth).

“In turn, knowledge of the data in a small cylinder determines the values of all of its derivatives on N = { ( x, y ) : y = 0 } (if the data are smooth).”

Specification of a possibly large but finite number of derivatives does not lead to unique solutions, even when the constraint is imposed and the resulting solutions are globally defined and smooth.

“This contrasts to the case discussed in section 4, in which it is shown that specification of a possibly large but finite number of derivatives does not lead to unique solutions, even when the constraint is imposed and the resulting solutions are globally defined and smooth.”

Given the values of a C2 solution u(x,y) to (1.2) in the space-time ellipsoid Wε, one may slice it with a hyperplane containing the x-coordinate axes but otherwise arbitrarily oriented in y to determine a possible Zε, which in turn determines the solution over the larger conical region Rε with base Zε.

“Namely, given the values of a C 2 solution u ( x, y ) to (1.2) in the space-time ellipsoid”

• Also stated in: § 5. A variant of a uniqueness theorem of Courant · p. 21 · hedged

  “we may slice it with a hyperplane which contains the x -coordinate axes but which is otherwise arbitrarily oriented in y , to determine a possible Z ε , which in turn determines the solution over the larger conical region R ε with base Z ε .”