From H^{1} to L^{2}

Characteristic boundary value problems: estimates from H1 to L2

Abstract.

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space , can be transformed into an a priori estimate of the same problem.

Key words and phrases:
Boundary value problem, characteristic boundary, pseudo-differential operators, anisotropic and conormal Sobolev spaces, Magneto-Hydrodynamics.
2000 Mathematics Subject Classification:
35L40, 35L50, 35L45.

1. Introduction and main results

The present paper is motivated by the study of certain non linear free boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics (MHD).

The well-posedness of initial boundary value problems for hyperbolic PDEs was studied by Kreiss [15] for systems and Sakamoto [28, 29] for wave equations. The theory was extended to free-boundary problems for a discontinuity by Majda [17, 18]. He related the discontinuity problem to a half-space problem by adding a new variable that describes the displacement of the discontinuity, and making a change of independent variables that “flattens”the discontinuity front. The result is a system of hyperbolic PDEs that is coupled with an equation for the displacement of the discontinuity. Majda formulated analogs of the Lopatinskiĭ and uniform Lopatinskiĭ conditions for discontinuity problems, and proved a short-time, nonlinear existence and stability result for Lax shocks in solutions of hyperbolic conservation laws that satisfy the uniform Lopatinskiĭ condition (see [3, 19] for further discussion).

Interesting and challenging problems arise when the discontinuity is weakly but not strongly stable, i.e. the Lopatinskiĭ condition only holds in weak form, because surface waves propagate along the discontinuity, see [11, 14]. A general theory for the evolution of such weakly stable discontinuities is lacking.

A typical difficulty in the analysis of weakly stable problems is the loss of regularity in the a priori estimates of solutions. Short-time existence results have been obtained for various weakly stable nonlinear problems, typically by the use of a Nash-Moser scheme to compensate for the loss of derivatives in the linearized energy estimates, see [2, 7, 12, 32, 34].

A fundamental part of the general approach described above is given by the proof of the well-posedness of the linear boundary value problems (shortly written BVPs in the sequel) obtained from linearizing the nonlinear problem (in the new independent variables with “flattened”boundary) around a suitable basic state. This requires two things: the proof of a linearized energy estimate, and the existence of the solution to the linearized problem.

In case of certain problems arising in MHD, a spectral analysis of the linearized equations, as required by the Kreiss-Lopatinskiĭ theory, seems very hard to be obtained because of big algebraic difficulties. An alternative approach for the proof of the linearized a priori estimate is the energy method. This method has been applied successfully to the linearized MHD problems by Trakhinin (see [33, 35] and other references); typically the method gives an a priori estimate for the solution in the conormal Sobolev space (see Section 2 for the definition of this space) bounded by the norm of the source term in the same function space (or a space of higher order in case of loss of regularity).

Once given the a priori estimate, the next point requires the proof of the existence of the solution to the linearized problem. Here one finds a new difficulty. The classical duality method for the existence of a weak solution requires an a priori estimate for the dual problem (usually similar to the given linearized problem) of the form (from the data in the interior to the solution, disregarding for simplicity the boundary regularity). In case of loss of derivatives, when for the problem it is given an estimate of the form , one would need an estimate of the form for the dual problem, see [10].

The existence of a solution directly in would require an a priori estimate for the dual problem in the dual spaces (possibly of the form in case of loss of regularity), but it is not clear how to get it.

This difficulty motivates the present paper. We show that an a priori estimate of the solution to certain BVPs in the conormal Sobolev space can be transformed into an a priori estimate, with the consequence that the existence of a weak solution can be obtained by the classical duality argument.

The most of the paper is devoted to the proof of this result. In the Appendix we present some examples of free-boundary problems in MHD that fit in the general formulation described below.

For a given integer , let denote the dimensional positive half-space

We also use the notation . The boundary of will be sistematically identified with .
We are interested in a boundary value problem of the following form

(1a)
(1b)

In (1a), is the first-order linear partial differential operator

(2)

where the shortcut , for , is used hereafter and denotes the identity matrix. The coefficients , () are real matrix-valued functions in , the space of restrictions to of functions of 1.
In (1b),

(3a)
(3b)

are first-order linear partial differential operators, acting on the tangential variables ; for a given integer , the coefficients , and , (for ) are functions in taking values in the spaces and respectively. Finally, in (1a) and , in (1b) stand for “lower order operators” of pseudo-differential type, acting “tangentially” on , whose symbols belong to suitable symbol classes introduced in Section 3.1. The operators , , must be understood as some “lower order perturbations” of the leading operators , and in the equations (1); in the following we assume that the problem (1), with given operators , , , obeys a suitable a priori estimate which has to be “stable” under the addition of arbitrary lower order terms , , in the interior equation (1a) and the boundary condition (1b) (see the assumptions , below). The structure of the operators (3) and , , in (1) will be better described later on.
The unknown , as well as the source term , are valued functions of , the unknown is a scalar function of 2 and the boundary datum is an valued function of . We may assume that and are compactly supported in the unitary dimensional positive half-cylinder . Analogously, we assume that and are compactly supported in the unitary dimensional ball . For an arbitrary , we also set and .
The BVP (1) has characteristic boundary of constant multiplicity in the following sense: the coefficient of the normal derivative in displays the block-wise structure

(4)

where , , , are respectively , , , sub-matrices, such that

(5)

and is invertible over . According to the representation above, we split the unknown as ; and are said to be respectively the noncharacteristic and the characteristic components of .
Concerning the boundary condition (1b), we firstly assume that the number of scalar boundary conditions obeys the assumption . As regards to the structure of the boundary operator in (3b), we require that actually it acts nontrivially only on the noncharacteristic component of ; moreover we assume that the first-order leading part of only applies to a subset of components of the non characteristic vector , namely there exists an integer , with , such that the coefficients , of take the form

(6)

where the matrices () and belong respectively to and . If we set , then the operator in (3b) may be rewritten, according to (6), as

(7)

where is the first-order leading operator

(8)

As we just said, the operator must be understood as a lower order perturbation of the leading part of the boundary operator in (7); hence, according to the form of , we assume that only acts on the component of the unknown vector , that is

(9)

A BVP of the form (1), under the structural assumptions (4)-(7), comes from the study of certain non linear free boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-hydrodynamics. Such problems model the motion of a compressible inviscid fluid, under the action of a magnetic field, when the fluid may develop discontinuities along a moving unknown characteristic hypersurface. As we already said, to show the local-in-time existence of such a kind of piecewise discontinuous flows, the classical approach consists, firstly, of reducing the original free boundary problem to a BVP set on a fixed domain, performing a nonlinear change of coordinates that sends the front of the physical discontinuities into a fixed hyperplane of the space-time domain. Then, one starts to consider the well posedness of the linear BVP obtained from linearizing the found nonlinear BVP around a basic state provided by a particular solution. The resulting linear problem displays the structure of the problem (1), where the unknown represents the set of physical variables involved in the model, while the unknown encodes the moving discontinuity front. The solvability of the linear BVP firstly requires that a suitable a priori estimate can be attached to the problem.
Let the operators , , be given, with structure described by formulas (2), (3a), (4)-(8) above. We assume that the two alternative hypotheses are satisfied:

  • A priori estimate with loss of regularity in the interior term. For all symbols , and , taking values respectively in , and , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8) and a finite number of semi-norms of , , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

    (10)

    where , and , , are respectively the pseudo-differential operators with symbols , , .

  • A priori estimate without loss of regularity in the interior term. For all symbols and , taking values respectively in and , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8) and a finite number of semi-norms of , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

    (11)

    where , and , are respectively the pseudo-differential operators with symbols , .

The symbol class and the related pseudo-differential operators will be introduced in Section 3.1. The function spaces and the norms involved in the estimates (10), (11) will be described in Section 2.
By the hypotheses and , we require that an a priori estimate in the tangential Sobolev space (see the next Section 2 and Definition 3 below) is enjoyed by the BVP (1). The structure of the estimate is justified by the physical models that we plan to cover (see the Appendix B). The inserting of the zeroth order terms , , in the interior source term and the boundary datum is a property of stability of the estimates (10), (11), under lower order operators. We notice, in particular, that the addition of in the boundary condition (1b) only modifies the zeroth order term for the part that applies to the components of the noncharacteristic unknown vector , see (7), (9). This behavior of the boundary condition, under lower order perturbations, is inspired by the physical problems to which we address. It happens sometimes that the specific structure of some coefficients involved in the zeroth order part of the original ”unperturbed” boundary operator (7) is needed in order to derive an a priori estimate of the type (10) or (11) for the corresponding BVP (1); hence these coefficients of the boundary operator must be kept unchanged by the addition of some lower order perturbations.
Note also that the two a priori estimates in (10), (11) exhibit a different behavior with respect to the interior data: in (10) a loss of one tangential derivative from the interior data occurs, whereas in (11) no loss of interior regularity is assumed. According to this different behavior, a stability assumption under lower order perturbations of the interior operator is only required in .
Both the estimates exhibit the same loss of regularity from the boundary data.

The aim of this paper is to prove the following result.

Theorem 1.

Assume that the operators , , have the structure described in (2), (3a), (4)-(8). Let .

  • If the assumption holds true, then for all symbols , there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8), and a finite number of semi-norms of , , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

    (12)

    where and .

  • If the assumption holds true, then for every pair of symbols there exist constants , , depending only on the matrices , , , , , in (2), (3), (7), (8), and a finite number of semi-norms of , , such that for all functions , compactly supported on , , compactly supported on , and all the following a priori energy estimate is satisfied

    (13)

    where and .

The paper is organized as follows. In Section 2 we introduce the function spaces to be used in the following and the main related notations. In Section 3 we collect some technical tools, and the basic concerned results, that will be useful for the proof of Theorem 1, given in Section 4.
The Appendix A contains the proof of the most of the technical results used in Section 4. The Appendix B is devoted to present some free boundary problems in MHD, that can be stated within the general framework developed in the paper.

2. Function Spaces

The purpose of this Section is to introduce the main function spaces to be used in the following and collect their basic properties. For and , we set

(14)

and, in particular, .
The Sobolev space of order in is defined to be the set of all tempered distributions such that , being the Fourier transform of . For , the Sobolev space of order reduces to the set of all functions such that , for all multi-indices with , where we have set

and , as it is usual.
Throughout the paper, for real , will denote the Sobolev space of order , equipped with the depending norm defined by

(15)

( are the dual Fourier variables of ). The norms defined by (15), with different values of the parameter , are equivalent each other. For we set for brevity (and, accordingly, ).
It is clear that, for , the norm in (15) turns out to be equivalent, uniformly with respect to , to the norm defined by

(16)

Another useful remark about the parameter depending norms defined in (15) is provided by the following counterpart of the usual Sobolev imbedding inequality

(17)

for arbitrary and .

Remark 2.

In Section 4, the ordinary Sobolev spaces, endowed with the weighted norms above, will be considered in (interpreted as the boundary of the half-space ) and used to measure the smoothness of functions on the boundary; regardless of the different dimension, the same notations and conventions as before will be used there.

The appropriate functional setting where one measures the internal smoothness of solutions to characteristic problems is provided by the anisotropic Sobolev spaces introduced by Shuxing Chen [8] and Yanagisawa, Matsumura [36], see also [30] . Indeed these spaces take account of the loss of normal regularity with respect to the boundary that usually occurs for characteristic problems.
Let be a monotone increasing function such that in a neighborhood of the origin and for any large enough.
For , we set

Then, for every multi-index , the differential operator in the tangential direction (conormal derivative) of order is defined by

Given an integer the anisotropic Sobolev space of order is defined as the set of functions such that , for all multi-indices and with , see [22] and the references therein. Agreeing with the notations set for the usual Sobolev spaces, for , will denote the anisotropic space of order equipped with the depending norm

(18)

Similarly, the conormal Sobolev space of order is defined to be the set of functions such that , for all multi-indices with . For , denotes the conormal space of order equipped with the depending norm

(19)

In the end, we remark that the following identity holds true. However, for a Sobolev order the continuous imbedding is fulfilled with the strict inclusion relation.
Since the functions we are dealing with, throughout the paper, vanish for large (as they are compactly supported on ), without the loss of generality we assume the conormal derivative to coincide with the differential operator from now on 3. This reduction will make easier to implement on conormal spaces the technical machinery that will be introduced in the next Section.

3. Preliminaries and technical tools

We start by recalling the definition of two operators and , introduced by Nishitani and Takayama in [26], with the main property of mapping isometrically square integrable (resp. essentially bounded) functions over the half-space onto square integrable (resp. essentially bounded) functions over the full space .
The mappings and are respectively defined by

(20)

They are both norm preserving bijections.
It is also useful to notice that the above operators can be extended to the set of Schwartz distributions in . It is easily seen that both and are topological isomorphisms of the space of test functions in (resp. ) onto the space of test functions in (resp. ). Therefore, a standard duality argument leads to define and on , by setting for every

(21)
(22)

( is used to denote the duality pairing between distributions and test functions either in the half-space or the full space ). In the right-hand sides of (21), (22), is just the inverse operator of , that is

(23)

while the operator is defined by

(24)

for functions . The operators and arise by explicitly calculating the formal adjoints of and respectively.
Of course, one has that ; moreover the following relations can be easily verified (cf. [26])

(25)
(26)
(27)
(28)

whenever and (in (25) and are even allowed).
From formulas (27), (28) and the boundedness of , it also follows that

(29)

is a topological isomorphism, for each integer and real .
The previous remarks give a natural way to extend the definition of the conormal spaces on to an arbitrary real order . More precisely we give the following

Definition 3.

For and , the space is defined as

and is provided with the norm

(30)

It is obvious that, like for the real order usual Sobolev spaces, is a Banach space for every real ; furthermore, the above definition reduces to the one given in Section 2 when is a positive integer. Finally, for all , the operator becomes a topological isomorphism of onto .
In the end, we observe that the following

are linear continuous operators, where denotes the Schwartz space of rapidly decreasing functions in and the space of infinitely smooth functions in , with bounded derivatives of all orders; notice also that the last maps are not onto. Finally, we remark that

(31)

is a bounded operator.

3.1. A class of conormal operators

The operator, defined at the beginning of Section 3, can be used to allow pseudo-differential operators in acting conormally on functions only defined over the positive half-space . Then the standard machinery of pseudo-differential calculus (in the parameter depending version introduced in [1], [6]) can be re-arranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to derive from the estimate (10) or (11) associated to the BVP (1) the corresponding estimate (12) or (13) of Theorem 1.
Let us introduce the pseudo-differential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [9].

Definition 4.

A parameter-depending pseudo-differential symbol of order is a real (or complex)-valued measurable function on , such that is with respect to and and for all multi-indices there exists a positive constant satisfying:

(32)

for all and .

The same definition as above extends to functions taking values in the space (resp. ) of real (resp. complex)-valued matrices, for all integers (where the module is replaced in (32) by any equivalent norm in (resp. )). We denote by the set of depending symbols of order (the same notation being used for both scalar or matrix-valued symbols). is equipped with the obvious norms

(33)

which turn it into a Fréchet space. For all , with , the continuous imbedding can be easily proven.
For all , the function is of course a (scalar-valued) symbol in .
Any symbol defines a pseudo-differential operator on the Schwartz space , by the standard formula