Smoothing Solutions to Initial-Boundary Problems for First-Order Hyperbolic Systems
We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain distributions of any order of singularity. It is known that such problems have a unique continuous solution if the initial data are continuous. In the case of strongly singular initial data we prove the existence of a (unique) delta wave solution. In both cases, we say that a solution is smoothing if it eventually becomes -times continuously differentiable for each . Our main result is a criterion allowing us to determine whether or not the solution is smoothing. In particular, we prove a rather general smoothingness result in the case of classical boundary conditions.
Solutions to hyperbolic PDEs demonstrate a wide spectrum of regularity behavior. The appearance of singularities in nonlinear cases is known as the blow up of a solution [1, 2]. Singularities can appear in a finite time even for small and smooth initial data . In some cases, both linear and nonlinear, a solution with time either preserves the same regularity as it has on the boundary or becomes less or more regular in time. The singularities encountered in the latter case are called anomalous [4, 5]. Criteria for the appearance of anomalous singularities are given in [6, 7, 8, 5, 9]. These papers are devoted to the case of interaction between singularities weaker than the Dirac measure. The singularities resulting from this interaction turn out to be weaker than the incoming singularities. A different effect is observed in [11, 12], where the incoming singularities are derivatives of the Dirac measure. In this case the interaction produces singularities stronger than the initial ones. We will focus on the phenomenon of improving regularity in the case of initial-boundary value problems with nonlinear local and nonlocal boundary conditions for first-order linear strictly hyperbolic systems.
Specifically, in the domain we address the problem
where , , and are real -vectors, , , and
Note that boundary conditions (3) cover the cases of classical boundary conditions (if do not depend on ) and reflection boundary conditions of local and nonlocal type.
We assume that
for all . Condition (5) occurs in many applications, where the functions for (resp. ) describe the “species” that travel to the left (resp. to the right) and are reflected in (resp. ) according to the boundary conditions (3).
We will impose the following smoothness assumptions on the initial data: The entries of , , , and are smooth in all their arguments in the respective domains, while the entries of are allowed to be either continuous functions or strongly singular distributions.
In the case of a continuous (considered in Section 2), by a solution to problem (1)–(3) we mean a continuous solution, i.e., a continuous vector-function in satisfying an integral system equivalent to (1)–(3). The existence and uniqueness of a continuous solution is proved in  (see Theorem 2). In the case of a strongly singular (considered in Section 4) by a solution we mean a delta wave solution, i.e., a weak limit of solutions to the original problem with regularized initial data that does not depend on a particular regularization. We refer the reader to [10, 9] for a more detailed definition and motivation of delta waves. Theorem 21 in Section 4 establishes the existence of a delta wave solution for a version of (1)–(3).
It is clear that the regularity of initial conditions (2) constraints the regularity of a solution if the latter is considered in the entire domain . However, the influence of the initial data can be suppressed if the regularity behavior is considered in dynamics, starting from a point of time .
Our main result is a smoothingness criterion for solutions to problem (1)–(3) in terms of the layout of characteristic curves (Theorems 12 and 22). In the case of classical boundary condition, the criterion implies that the solution is smoothing whenever for all . As another consequence, we obtain a class of boundary conditions under which the wave equation has smoothing solutions (see  for a special case of this result).
Our analysis of problem (1)–(3) shows a phenomenon usually observed in the situations when solutions to hyperbolic PDEs change their regularity: the smoothness changes jump-like rather than gradually. Another feature of problem (1)–(3) shown in  is that, even if we allow non-Lipschitz nonlinearities in (3), this system demonstrates almost linear behavior. The smoothingness effect in the non-Lipschitz setting contrasts with blowups [1, 2] observed in many nonlinear systems.
In [6, 7], results similar to ours are obtained in some more special cases, namely for homogeneous linear boundary conditions of local type with constant coefficients and continuous initial data. Some restrictions on (1)–(3) are imposed in [6, 7] by technical reasons, as the authors use an approach based on the Laplace transformation and the Green’s function method. We extend these results to the case of general linear first-order hyperbolic systems, nonlinear nonlocal boundary conditions, and distributional initial data of the Dirac delta type and derivatives thereof. Note that we use a different approach based on the classical method of characteristics. This method suits well for understanding of the mechanism of the smoothingness effect.
An essential technical difficulty to overcome in
demonstrating the regularity self-improvement is caused by the fact that
the domain of influence of initial conditions (2) is in general
infinite. In other words, the regularity of solutions all the time depends on the
regularity of the initial data. However, this dependence is different for the
boundary and the integral parts of the equivalent integral form
of problem (1)–(3). Since the boundary
summands are compositions of boundary data with functions defining characteristic curves
and hence are “responsible” for propagation of singularities, the smoothingness effect is encountered whenever
the boundary summands have a “bounded memory” or, more rigorously,
all characteristics of (1) are bounded and
each boundary singularity expands inside
along a finite number of characteristic curves.
The mathematical motivation of the paper is the scope of the stability theory, the Hopf bifurcation analysis, and the investigation of small periodic forcing of stationary solutions of hyperbolic PDEs. The main reason why those techniques are well established for nonlinear ODEs and parabolic PDEs, but not for nonlinear hyperbolic PDEs, is that, in contrast to parabolic case, hyperbolic operators in general do not improve the regularity of their solutions in time (the question is closely related to propagation of singularities along characteristic curves). This complicates, in particular, proving the Fredholmness property of the linearizations which is crucial for the analysis of solutions to nonlinear problems. We provide a range of boundary conditions ensuring the desired smoothingness effect, which still makes possible to handle the bifurcation analysis of a class of nonlinear problems (this idea goes back to ).
The practical motivation is caused by applications to mathematical biology , chemical kinetics (describing mass transition in terms of convective diffusion and chemical reaction and analysis of chemical processes in counterflow chemical reactors [17, 18, 19, 20]), and semiconductor laser dynamics (describing the appearance of self-pulsations of lasers and modulation of stationary laser states by time periodic electric pumping [21, 22, 23]).
2 Continuous initial data
where By we denote the Euclidian norm in .
Theorem 2 ([13, Thm 3.1])
Assume that the data , , , , and are continuous functions in all their arguments, and the coefficients are Lipschitz in locally in . Suppose that are continuously differentiable in and for each there exists such that
where is a polynomial in with coefficients in . If the zero-order compatibility conditions (6) are fulfilled, then problem (1)–(3) has a unique continuous solution in which can be found by the sequential approximation method.
We now introduce the notions of an Expansion Path and an Influence Path, that will be our main technical tools. Let denote the characteristic of the -th equation of (1) passing through . Let be a characteristic of the -th equation of system (1). Suppose that reaches at two points. Let be that of these points having larger ordinate (hence, or ). We say that reflects at if
In this case that of the characteristics and which lies above the line is called a reflection of . If the reflection is defined by , it will be called a jumping reflection.
Note that condition (8) means that the -th components of the vector participates in evaluation of for and of for . Whenever the continuous solution to problem (1)–(3) is known (see Theorem 2), condition (8) is easily checkable. Otherwise, one can use the following constructive sufficient condition: (8) holds true whenever for every there is such that
A sequence of characteristics is called an Extension Path (EP) if each is a reflection of (see Figure 1).
A sequence of curved segments is called an Influence Path (IP) if the following conditions are met.
Each is a continuous part of a characteristic of the -th equation for some ;
The whole path is monotone in the sense that the coordinate continuously increases while moving along it;
The transition from to can be of three types:
and meet at a point such that in any neighborhood of ;
and meet at a point with or and in any neighborhood of a characteristic of the -th equation reflects to a characteristic of the -th equation;
terminates at a point with or , starts at the point on the opposite side of , and in any neighborhood of a characteristic of the -th equation makes a jumping reflection to a characteristic of the -th equation.
Roughly speaking, an IP is a piecewise continuous curve with smooth peaces lying on characteristic curves such that either and meet within or terminates at and starts at the point on the opposite side of .
Note that any EP is an IP, while the opposite is not necessary true because segments of an IP not necessary lie on reflected characteristics.
Define a set , called the domain of influence of the initial data on , as follows: if the value can be changed by varying a function in (2).
if and only if there is an IP emanating from the initial axis and going through a part of the characteristic .
The sufficiency follows from the proof of the necessity in Theorem 12. The proof of the necessity is based on the constructive description of the domain of dependence of at which turns out to be the union of the IPs going through a part of below the line . Under the domain of dependence of at we mean the set of all points such that if varying the function
Define a set to be the union of IPs emanating from the initial axis and satisfying Definition 5 with additional conditions imposed on the three transition types from to :
and meet at a point such that ;
and meet at a point with or and at this point reflects to ;
terminates at a point with or , starts at the point on the opposite side of , and at the characteristic makes a jumping reflection to .
Note that .
We now introduce two conditions that will occur in formulation of our results.
() For every there exists such that, for all , every EP passing through lies below the line .
() For every and there exists such that, for all with , every EP containing the characteristic lies below the line .
In many important cases conditions () and () can easily be reformulated and verified in terms of , , and . Sometimes they can be verified even directly (see examples in Section 3).
Let . In the domain let us consider problem (1), (9), (3) with for all (i.e., system (1) is decoupled) and with replaced by in (9). Then condition means that, whatever and , the function has a bounded domain of influence on for every . In other words, for any decoupled system (1), if is singular at some point , then this singularity expands outside along a finite number of characteristic curves within for some that does not depend on . In contrast to , condition means that, whatever and , the function restricted to has a bounded domain of influence on for every .
One can easily see that the sufficient condition () implies the necessary condition (), while the converse is not true. Nevertheless, in a quite general situation when for every , it is not difficult to observe that conditions and are equivalent. Hence we have the following result.
Proof. Sufficiency. Assume that condition () is fulfilled. Define a sequence inductively by the following rule. Let be the infimum of those for which there is and an EP passing through and lying below the line ; let for be the infimum of those for which there is and an EP passing through and lying below the line . Note that is monotone and approaches the infinity. The latter fact is a simple consequence of the smoothness assumptions on . By Theorem 2, problem (1)–(3) has a unique continuous solution in . It suffices to show that .
where and denotes the smallest value of at which the characteristic reaches . In the sequel, along with the equation we will also use its inverse form . Due to the definition of , in the boundary term in (10) we can substitute
where if and if . Continuing in this fashion, the right-hand side of (10) can eventually be brought into a form depending neither on nor on . This version of will be referred to as . We begin with establishing the -smoothness of . It will be proved once we show that the right-hand side of has a continuous partial derivative in . The latter can be done by transforming (appropriate changing of variables in) all integrals occurring in . The transformation of each integral follows the same scheme, which we illustrate by example of the integral expression
We will use assumption (5). Suppose, for instance, that , , , and . This entails, in particular, that and .
Due to (10), we obtain (up to the sign)
where is the area shown in Fig. 2 and denotes the -coordinate of the point where the characteristics and intersect. The other cases are similar. For example, if , then in the formula (13) index should be replaced by and the integration over should be replaced by the integration over .
The desired -smoothness of follows from the -smoothness of . The latter is a consequence of the smoothness properties of . Indeed, from the equality we conclude that, if exists, then it is given by the formula
Thanks to the equality and condition (5), the function is continuously differentiable in .
Thus, the right-hand side of is continuously differentiable in . Therefore, . The membership of in now directly follows from system (1).
In the next step, we prove that . For we have equations
for , and for . Here denotes the scalar product in . In (15) we can represent in the form
We continue in this fashion up to getting a representation of the boundary term (15) that does not depend on , what is possible due to the definition of . To show that the right-hand side of the obtained expression for , say , is continuously differentiable in , we transform all integrals contributing into similarly to (13). Using the fact that , one can easily show that . The desired -smoothness of the solution is then a direct consequence of system (1) and its differentiations.
We further proceed by induction on . Assuming that for some , we will prove that . We differentiate (1) times in , thereby obtaining a system for . The integral form of this system is similar to . Analogously to , the definition of makes possible an integral representation of the system for which does not depend on and includes integrals of similar to . To show the -smoothness of the solution, we transform the integral terms analogously to (13). Finally, the -smoothness of outside of follows from suitable differentiations of system (1). The sufficiency is thereby proved.
Necessity. Suppose that condition () is not fulfilled and prove that the solution to problem (1)–(3) is not smoothing for some . Fix and such that for all there is an EP containing the characteristic for some and going beyond .
Since all singularities of solutions expand along EPs, it is sufficient to prove that there exist and such that, whatever , the solution is not -smooth at .
By Lemma 8, for any there is an Influence Path emanating from the line and going through a part of . Denote the smallest possible “length” of such a path by . By the smoothness assumptions on the initial data, if is sufficiently close to and , then . Since is closed, the standard compactness argument implies that is bounded by a constant uniformly over all .
Let be a continuous nowhere differentiable function. Consider an arbitrary . Let us fix a shortest Influence Path from some to with lying on and with the transition from to as in Definition 9. Using the smoothness assumptions on and , we can suppose that . We now intend to prove that the solution is not -smooth at . Since , this will give us the necessary part of the theorem. Let denote the starting point of . Suppose that is a part of a characteristic of the -th equation. As is not continuously differentiable at , the function is not -smooth along due to the definition of a characteristic. If is a reflection of , then is not differentiable at and hence is not differentiable along . Otherwise, by Definition 5, we have and is not -smooth along , because the integral of along in the integral form of the -th differential equation is not a -function and this nonsmoothness cannot be compensated by any other summands in the integral representation of our problem. It follows that is not -smooth, in particular, at . Continuing in this way, we arrive at the conclusion that is not -smooth along and hence is not -smooth at .
Going into the details of the above argument, let us represent in an integral form with an integration over a neighborhood of the Influence Path . We focus on the case of where characteristic is not a reflection of and is not a reflection of (see Fig. 3). A similar or even simpler argument works as well for other possible cases. Extend each to Given , let denote the union of and one-sided neighborhoods of , bounded by the characteristics and . Fix an so that contains neither nor .
Now we write an integral representation of in terms of over . We start from the formula (10) for and rewrite a part of the integral in the right-hand side as
By construction, . Furthermore, we consider the part of the integral in (16) over the area denoted in Fig. 3 by and transform it similarly to above as follows: